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Newbie Progression Guide
Post your e-mail in the thread so goons can send you a referral code! This will give you a 1 week exp booster!
Send an application to the GOON clan. The sooner you get in the easier your life will be.
PLEASE DO NOT SOLO THIS GAME IS NOT BALANCED AROUND SOLO PLAY
WEBSITES
Beyond this guide, it is important to be aware of these websites:
https://deathsnacks.com/wf/ - tracks alerts and invasions, also has a mobile app
http://warframe.market/ - a better market place then standing on a relay ingame and waiting for passersby
http://www.tennodrops.com/ - database of item drop locations
MARKET
The market is an almost unorganized mess in which many useful things are buried between tons of cosmetics or needlessly expensive "cheats" (crafting resources, boosters, other items). Make sure you check the market to get an overview which things can be bought for credits, or whose blueprints you can buy that don't require advanced materials (particularly important for weapons, sentinels).
Unless you have way too much money to spare, it's a good idea to be patient and/or stingy with buying plat, since you will most likely receive a 75% platinum discount voucher via daily rewards within the first few weeks. Also keep in mind that there's a thriving market for rare mods or prime parts, which you can sell to other players for plat.
If you're tempted to spend plat, ask your fellow goons first, and spend it only on things that are an absolutely pain to farm otherwise, or on cosmetic items.
MOD SYSTEM
Warframe has an unorthodox weapon modding/improvement system. You don't increase weapon parameters by leveling it, you instead increase its mod capacity. The mods themselves are what improve the weapon's performance/give it special effects. So a modless weapon at 30 is just as effective as a modless weapon at 0.
The tutorial gives you Damaged mods. Do not level these. Slap them on your weapons but replace them with the regular variety as soon as you can.
The important mods are Serration, Hornet Strike, Pressure Point. These straight up increase damage for your Primary, Secondary and melee respectively.
You will eventually want elemental damage for all your weapons, but when you're starting out, just slap on any mods which increase the damage you do.
Fury increases your melee attack speed.
Split Chamber and Barrel Diffusion are Multishot mods for rifle and pistols respectively. These let you fire more bullets without consuming additional ammo. In practical terms, this basically increases your weapon's attack power even further, so pick these up as soon as you are able too. These should be the last of the priority weapon mods since they're quite rare.
Shotgun mods have their own category. Point Blank is the damage increase mod and Hell's Chamber is the Multishot mod. Shotguns like the Hek are really good nowadays so make sure to pick one up when you're Mastery Rank 4.
For your warframe, pick up Continuity, Stretch, Streamline and Intensify. You may also want Redirection and Vitality to increase your shields and HP respectively, but these are very common.
WEAPON PROGRESSION
This is the recommended progression for newbies. I am using the weapons which allow you to coast along with minimal fuss. Note that past MR 3 these basically become a guideline, feel free to use whatever looks cool/can be afforded. I stop after MR 6 or so because by that point you should know the game well enough to figure the rest on your own.
If you see a weapon and it has a variant (they usually have a prefixed or suffixed name, like Simulor has a variant in Synoid Simulor, Boltor has variants in Boltor Prime and Telos Boltor and so on) it's recommended to level the vanilla version to 30 and then sell it and then use the variant. The variants always have better stats than the vanilla versions.
Weapons with a (GF) are Goon Favorites; recommended by myself and the rest of the clan. Keep these as they are usually very good well into the endgame.
Primary:
[MR 0] (Whichever weapon you picked in the tutorial) -> Braton (buy it immediately after) -> Karak
[MR 2] Boltor (Prime variant is GF)
[MR 3] Tetra, Gorgon
[MR 4] Ignis (GF), Tiberon, *
[MR 5] Sybaris, Tonkor (GF)
[MR 6] Soma (Prime variant is GF)
If you feel like messing with shotguns:
[MR 2] Kohm (GF)
[MR 3] Drakgoon (debatably GF)
[MR 4] Hek (GF, yes even though Vaykor Hek is a thing) , Tigris (The Sancti variant (Sancti Tigris [MR 12]) is GF)
* Quanta [MR 4] is a solid weapon which is accurate and ammo efficient but it has fairly high building requirements. Used to be GF but has fallen out of favor somewhat
Secondary:
[MR 0] (Whichever weapon you picked in the tutorial) -> (Most secondary weapons before MR 2 are trash so I will list the ones that you can get for cheap, level to 30 and then sell right after) Lato, Aklato, Lex, Aklex
[MR 1] Sicarus
[MR 2] Akstiletto, Gammacore**, Hikou, Kunai, *
[MR 3] Catanas
[MR 4] Angstrum, Atomos (GF)
[MR 5] Marelok
[MR 6] Synoid Gammacor (GF)**, Vaykor Merelok (GF)
* I highly recommend Sonicor (GF) [MR 2] but its ingredients may be hard to come by.
** Synoid Gammacor [MR 6] is a fantastic weapon, but opinions are divided due to its ammo consumption. Personally, its my secondary weapon of choice due to its damage, accuracy and innate proc effect.
Melee:
[MR 0] (Whichever weapon you picked in the tutorial) -> Cronus, Amphis, Heat Sword, Karyst (There are a shitton on this tier, pick whichever you can afford. Cronus and Heat Sword blueprints are given as a Vor boss drop and quest reward respectively though)
[MR 1] Magistar
[MR 2] Atterax, Scindo, Fragor, Orthos, *
[MR 3] Broken-War (GF)**, Anku, Galatine (GF)
[MR 4] Nikana***, Redeemer (GF)
[MR 5] Jat Kittag (GF)
[MR 6] [Melee MR 6 is a bleh tier]
[MR 7] Destreza (GF)
* Scindo, Fragor and Orthos have prime variants which are all (GF)
** Obtained for free from a quest. Comes potatoed too. Best sword currently in the game.
*** Nikana will be used to make the superior Dragon Nikana [MR 8]. There exists Nikana Prime (GF) which is better than both Nik and DNik, but its parts are rare. DNik is a solid weapon though.
Special Note: I must also mention War (GF) [MR 10]. This is currently the most powerful weapon in the game by raw damage alone. But getting it is a complete and utter pain in the dick and requires you to use up a Broken-War. While the OP explicitly discourages dropping plat on most things, I for one, recommend dropping plat on the Hunhow's Gift bundle. In addition to getting a lot of neat skins, you also get War, fully potatoed and with its own slot. It costs 520 platinum, but you can buy it at any time and bypass both the grind and the rank requirement.
Warframe
(Whichever frame you picked in the tutorial) -> Rhino [MR 2] -> Whatever else you want
Rhino is the only frame that requires a certain Mastery Rank to unlock and annoyingly it's one of the most useful to a newbie. His Iron Skin greatly increases your survivability and he is a fantastic newbie frame. Ask clan members to taxi you to the places where the resources needed for Rhino drop: Void (Control Module), Jupiter (Neural Sensor), Saturn/Phobos (Plastids)
Beyond Rhino, use whatever sounds cool.
Companions
Pick the Carrier up ASAP! You may wanna just get it and its weapon to 30 and then replacing it with the Carrier Prime. Regardless, make sure you have it, as it increases the range you can pick-up items, the single biggest QoL change in the game. It is absolutely vital for resource collection and provides a lot of passive quality of life features. The other Sentinels are sadly not as useful, many have useful weapons or effects, but none are as much of a game changer as the Carrier.
Pick up its blueprints on the market for 100k and then gather the materials and build it. Ask goons for assistance. Get it. GET IT. GET IT!
Don't bother with Kubrows or Kavats (dogs and cats respectively) until you get a decent cash flow! The quest to get a kubrow requires a 100k blueprint to build an item that also costs 100k to get a companion that requires 65k worth of upkeep. That's huge. Put it off till way later.
GAME MECHANICS
Various parts of the game are locked behind progression gates and/or quests, as follows is roughly a list of places to go and things to do to unlock parts of the game.
Sanctuary
Found on [Mercury] in the Larunda Relay. Allows training for mastery tests, sells specific items in exchange for scans of enemies (scanner upgrades, some mods, Simulacrum, Exilus).
First go to the navigation screen and select Mercury, then the Larunda Relay.
Once that has loaded, open your menu and click the Conclave option.
Once there, close the screen that's open.
Then turn to the right, and walk into the tube going upwards.
In the tube, hit X in the middle to go up, or just climb up.
Topside, go through the door with the crown on it.
Once inside, hang a direct right, and there are the terminals for the mastery tests.
Star Char Progression
Syndicates
Unlocks at [MR 3]. Sell various items, but most importantly syndicate weapons and warframe ability augmentation mods.
Relics
Given by [Mars Junction]. Required to craft Prime items.
Sorties
Unlocks at [MR 4]. High level missions with rewards unattainable otherwise. Bring Rhino, other frames only if you're confident you can take them on.
Simulacrum
Sold in [Sanctuary]. Training area which allows spawning enemies.
I'm getting the hang of Warframe!
Excellent. Let's move to some advanced topics.
FORMA AND YOU
Got a weapon you particularly like but you need more mod cost than your item's mod point total allows? Get the item to 30 and polarize a slot with a Forma. This will permanently add said polarized slot to that weapon (as well as allowing you to switch polarity slots with each other). This will reset the weapon to level 0 so you have to level it to 30 again. If you still don't have enough mod points to fit all the mods you want, repeat the process.
You get Forma BPs nowdays from Relics. Complete the Earth-Mars junction and collect some relics that have Forma BPs on their drop table. Run the Void Fissure mission to open the relic and hopefully you'll get a Forma BP. Then make one (has a 24 hour build time, mind) and use it. Due to their long build times, and since you need like a million of the damn things, it's recommended you have a backlog of Forma BPs so you can always keep building Forma. Never not build Forma.
ARCHWING
Archwing (usually called AW) is the game's space flight mode unlocked by a quest at MR2. It is a...polarizing game mode. Archwing currently features six degrees of freedom and lets you fly in space or spaceship trenches to do missions.
Your default equipment is okay. The Odonata comes in a Prime variant, but frankly both are outclassed by other options.
You can pick up the Amesha from the dojo. This is an AW that is incredibly tanky and is built for survivability. If you find yourself having trouble in AW, get an Amesha!
The Itzal is another favorite found in the dojo. This is more of a "rogue" AW, you can use it to go invisible and zip around instantly but it has low defenses. It is nevertheless an excellent AW and is probably the most fun of the bunch. Once you get the hang of flying, this is probably gonna be your go-to AW. It also lets you summon drones and at max power str you can have like 10 of em all lasering an enemy at once 
The Elytron is less versatile as it is slower and more focused on stationary defensive objectives (like in mobdef and int) but is still fantastic due to its power and range. They are all resource-heavy unfortunately.
Your default Archgun, the Imperator, is pretty good as is. Side upgrades include Grattler and the Cyngus (yeah Cyngus, not Cygnus). The latter is easier to get, ask goons to grab parts from syndicates.
As for Archmelee, replace the Veritux ASAP. Better options include the Knux, the Prisma Veritux (If Baro is around) and especially, ESPECIALLY the CENTAUR.
EXTRACTORS
Many people do not use these and I have no idea why. Pick up the Extractor Blueprint from the market, these come in two flavors: Titan and Distilling. Titans collect less resources but they pick them up quickly, while Distilling Extractors can potentially get more and rarer resources, but they take twice as long to dig. I recommend starting with Titan and moving to Distilling when you're in the double digit MRs.
You can place extractors on any planet whose nodes you cleared. I recommend clearing Mars and Earth ASAP (fair warning, they have AW missions and Earth's boss requires MR 5), these will help you get a lot of the necessary early game resources really quickly. MArs even moreso than Earth. Uranus is good too for Plastids and Polymers. Saturn and Jupiter are also good for Orokin Cells and Neural Sensors respectively, though bear in mind actually getting them from an extractor is very rare. Still, it's a zero effort way of getting more resources, so there's no reason not to.
When you place an extractor on a planet, it may take some damage. It recovers health while in your inventory and not in use. So have 2 Extractors per planet and swap between them. DO NOT PUT ANY EXTRACTOR WITH LESS THAN 100% HP ON A PLANET. If they go to 0, you will need to make a new extractor which costs valuable resources.
How many Extractors you can deploy depends on your Mastery rank. 0-4 lets you deploy 1 drone, 5-9 lets you deploy 2 and 10+ lets you deploy 3. If you are a founder, that's a +1 drone on top of that. If you ever bough Prime Access, that's yet another +1.
MYSTERIOUS E-MAILS
You may get messages from a dude when you kill bosses. Ignore it, it's probably nothing.
SECOND DREAM
One of the most important quests in the game is the Second Dream. You need to be at MR3 and have access to Uranus' junction to unlock it. First, you need to clear Natah. To get this quest, scan an Occulyst on Uranus. These guys appear randomly in any non-defense/interception map (use capture or exterminate to make your life easier). The more nodes you complete on Uranus, the higher the chances of getting them to come up. When you see the flash, just continue to your objective and you should run into them. You can use either Codex or Synthesis scanner, it doesn't matter.
At the end of the quest, pick Zenurik. If you are into melee, pick Naramon. Vazarin is decent too. Madurai is not great and Unairu is total trash. You can use these lenses by slotting them into your level 30 weapons and warframes (same way you would slot a potato). These will take a portion of your exp and convert it into focus points for the lens' school. You can use a Greater Lens to increase the conversion rate.
EXILUS
Exilus Adapters unlock the Exilus slot for Warframes. These can hold utility mods. Nowadays people mostly use it for the mods you get from Lua tests, particularly Power Drift and Cunning Drift (Coaction Drift has some fans too). You get these adapters from quests and you can build your own by buying the blueprint from Semaris for 50k reputation, though fair warning, they require 1 Argon Crystal, 1 Orokin Cell and 2 Forma to craft. You can buy em from the market for 20 p as well.