22Wrms is fine for most speakers. I powered a set of pioneer 3-ways (60Wrms) off 22rms for 6 years, and they sounded great. I could get them way louder than I needed, and I didn't notice any clarity issues.
OP, yes, most all aftermarket HU's have a built in amp. Good ones also have "pre-amp outputs" which give you a low wattage output line that you would connect to an amp. That way you aren't sending an amplified signal to another amp and multiplying the noise.
As you can see, there are a lot of opinions on this stuff. I'd highly recommend keeping it simple at first, and getting stuff that you can use in a better system later if you want. If you think you might want an amp in your system later, then be sure your HU has pre-amp outputs (i prefer 6 channel pre-amp outputs, which means you have front L/R, rear L/R, and subwoofer, that makes future integration of a sub and amps really easy).
Right now with limited funds, keep it to as few components as possible, and get decent components, which is why I recommend just getting a HU and front speakers. Yes it's easy enough to upgrade the rears to 6.5's, but why? It's not going to improve your sound quality much for the front seat, so you aren't getting alot of bang for your buck.
And I promise, you don't need an amp for your front speakers right off the bat, a HU is totally sufficient. If it sounds weak, then I'd recommend running new, better, thicker speaker wires to your doors. I did notice the difference when I did that. It's more of a PITA to run new wires, but it's cheap and effective.