'tis one of the best games around at the moment, and one of the best implementations of f2p.
The main thing to understand is that mods are where it's at - and fusing (upgrading) mods. After a while you'll get the hang of the "drain" system (mod costs). Roughly, what you want to aim for in the interim is:-
1) for the Warframe: Streamline, Flow, Intensify, Continuity and Stretch for the abilities (energy efficiency, big energy pool, power strength, duration, range). You'll see the effect of those mods on the percentages at the left pane in the Arsenal -> Upgrades page for your Warframe. For tank, you want Redirection (bigger shields) and Vitality (bigger health pool). If your frame also has high armor already, you also want Steel Fiber (e.g. Valkyr has almost no shields, big health pool and high armor, so for her you'd want to min-max Vitality and Steel Fiber). Another important "tanking" mod is Quick Thinking, which uses your energy pool as temp hp, but that's a rarer mod that comes later and is more useful in higher level content where you can be one-shotted by mobs.
Later on, you also start getting into the "corrupted" mods for the Warframes, which give a big boost to one thing with a fairly heavy cost to another. e.g. Fleeting Expertise bumps Efficiency much better than Streamline, at a severe cost to Duration - note that the efficiency cap is +75%, or 175%, so you only need 4/5 FE and 4/5 Streamline to hit that, or 3/5 FE and 5/5 Streamline for +70% with least Duration hit. Other mods like this are Overextended (+Range -Power), Narrow Minded (+Duration -Range), Transient Fortitude (+Power -Duration) and Blind Rage (+Power -Efficiency).
2) for Primary weapons, first priorities are Serration and Split Chamber. Then, if your weapon has 15% or more crit rating, you want Point Strike and Vital Sense (crit rating and multiplier). Then you want to fill out the "elemental" damage mods to taste, and there are mixed elemental/status mods to use if your weapon also has decent status (20% or above). The Shotgun has its own set of equivalents (in place of Serration, Point Blank, instead of Split Chamber, Hell's Chamber, and so on).
3) for Secondary weapons, you want Hornet Strike and the equivalent "multishot" mod to Split Chamber (forget its name). Thereafter, again, same min-maxing principle as with primary weapon: bump what the thing is already good at.
4) for melee, Pressure Point is the damage mod, and Fury is probably the closest equivalent to the "multishot" mods (it ups attack rate). Thereafter, again, same min-maxing principle.
Potato all the things you're using at first (i.e. you need an Orokin reactor, gold potato, in the frame, and an Orokin catalyst, silver potato, in the weapons), that will double the mod capacity. But then once you're in the saddle with your first few frames and a few weapons, you'll become more discriminating, and only potato things you love using/are actually powerful, and start to "forma" them (further increase mod capacity, resetting the weapon to 0, and building it up again with that higher mod capacity).
In parallel with this, play the market. Once you start getting Void Tower keys, start playing those missions solo a bit, and join in with T1D (Tower 1 Defence) missions when they come up in recruitment. They're not too hard, readily available, usually allow all comers (when tagged with "xp run") and can give a big boost to whatever frame/weapon you're levelling up. The more Void missions you do, the more Prime parts and better mods you start to accumulate, the more you can be self-sustaining by selling those bits and pieces enough to keep you in slots for new weapons and frames, and to pay for potatoes if you can't make them at the time (plus also of course accumulating and building Prime frames and weapons for yourself). Also you'll be able to afford to buy the occasional rarer mod if you can't be bothered farming for it (e.g. Fleeting Expertise is usually one of the first corrupted mods to get, and it's usually readily available on the player market for 10-20pl).
The game is great fun solo, but it's even more fun in teams. At the moment "coptering" is a thing to give little speed boosts to keep up with the team (crouch-jump-quick melee) but that's soon going to go.
Good luck, Tenno!