On the right are yourstats and skills (changes depending on what you mouse over).

On theleft are your current equips. Mouse over them to either change themby clicking equip, upgrade to equip mods and appearance to changecolors and equip alternate helmets (they give stat boosts as well asa new look).
Gear lets you use codex scanners, hp restores and ammo boxes. You can get these from the market and then equip them here by clicking on an empty panel and choosing your item. You must EQUIP gear before you can use them. This is where binding a key to gear slots helps or you will have to keep choosing which gear to use during the level.
For now go to your warframe and select upgrade, whichbrings us to:

The mods which youcollect that can be equipped will be on the left side. Next to thename of each mod is a number and symbol (polarity). The number showshow much mod capacity it will take up when equipped.

Notice that the total mod capacity you can equip is on the blue bar above the slots. The total mod capacity goes up by one for every level you gain on that equip, to a maximum of 30 (without orokin/potato).

Simply drag anddrop the mod you want to a slot on the right to equip it. You cannot equip two of the same mod on an equip.

The slotswith a pattern on the top right are polarity slots and are differentfor each warframe and weapon.

If you match thepolarity on the top right of the mods with a polarity slot then themod consumes half of what it states rounded up. However if you try toequip a different polarity then the mod consumption is increased by athird.

A difference between warframe and weapon slots are thatwarframes have an extra slot on the left. These are aura slots and ifyou get an aura card from an alert, you can equip it to increasemod capacity (match the polarity to double the increase!).
As your warframe is where you equip your skills (their mods are given automatically when you build your frame) and skills take up mod capacity, try and get any aura card in the beginning so that you will not run out of space just equipping skills.

Also notice on top youcan install an orokin reactor/catalyst(aka potato because of the shape) or polarize(with forma) yourgear. These are obtained by crafting from blueprints which are foundin alerts or can be bought from the market with platinum (realmoney).

Installing an orokin (aka potato) will double mod capacity (can only be done once per equip).

Polarizing with forma will let you change one of the slots into a polarity slot.This can be done multiple times but the level for the equip will dropback to zero. Leveling up these weapons again will not count towardsyour main mastery rank.

Once you are done equipping mods click apply and we'll now upgrade those mods.

Every level gained onyour equip gives you one more mod capacity. Click on mods on thebottom left

Click on a mod and youwill be given the 4 choices shown.

Transmute lets you combine 4 modsin order to get another random mod, I wouldn't recommend using thismethod though.

Fusion is how we level up mods. Choose a mod and clickfusion. Now click any mods you don't need and it will level up themod you first chose every time the bar on the left goes full. Each level up gives a good increase but alsoincreases mod consumption by one.
The best way to upgrade is to combine mods withduplicates and fusion cores (not used for anything else). Using different mods with different polarities to the one you want to upgrade will require four times more mods! However if you do not have any need for those mods then by all means go ahead.

If you can no longer see the fusion button that means the modis maxed.

Finally exit the modscreen and apply any changes you make.