Can you capture enemy planets?

You bomb it with a siege ship or capship, then colonise it with a colony frigate or colony capship.
I think it's explained in the tutorial. Use it.

Alternatively, you can build a culture producing structure (media hub/broadcast center/ temple of communion) on a nearby planet, and wait for their allegience to go to zero. This will cause the planet to become neutral, so that you can avoid (mostly) a pretty messy battle. The downside is that it takes so long to do, and you still need to send a colony ship to colonize the planet (which can also get pretty messy, since tactical structures don't disappear when the planer is neutralized).

Good luck!
ChaoticMagician

One last question. If I capture an enemy planet that still has buildings around it, do I get them?

No. They even have to be destroyed before you can build your own structures that you can use.