There are two ways of obtaining your next Warframe; an easy way and a more difficult way:
Easy method Purchase it with platinum. The advantage is that it comes with its own Warframe slot and pre-installed with either Orokin Reactors or Orokin Catalysts-- these double your mod capacity. The third advantage is that it’s instant. The downside of course is that you must spend real money.
Harder method Purchase the overall blueprint and obtain the blue prints to each piece. These are usually dropped by bosses. Once you have them, make the individual pieces (which can take hours). When you have the three pieces you can make the Warframe (you’ll also need an Orokin cell. This is a mat (material) and is not a Orokin Reactor or Orokin Catalyst. Making a Warframe takes another three days.
As enemies are killed, your character and any equipped gear will earn experience points-- whether you do the actual killing or not. You can play in groups to increase the amount of enemies that are killed, and to level up your equipment much faster. However, the specific equipment leveled appears to be determined by the type of weapon a teammate used. For instance, if your teammate uses a primary weapon, then your primary weapon is the one that will gain experience. Also worth noting is that you don't gain those points if you're not within a fairly limited range, so it pays to stick close to your friends to maximize your own benefits (and to enable teamwork, of course).