Terminology might depend on where you are. There isn't really any such thing as "regular yoga". I do know that in the US, many people associate "yoga" with hatha yoga, which usually means that the focus is on the building blocks of poses and on stretching-type poses.
Power yoga, on the other hand, is a vinyasa- or ashtanga-style, flow yoga; power yogis move quickly between poses, but the poses often focus on building strength rather than flexibility. (Any type of yoga will improve your flexibility; I'm speaking in relative terms here.) "Power yoga", while inspired by Ashtanga, does not follow a strict routine of poses. There's freedom for the poses to vary from one session to the next.
The usual recommendation regarding food is to wait at least 3-4 hours after eating a meal to practice any type of yoga, or 1-2 hours after a light snack. If you really need to eat right before doing yoga, have something light like a piece of fruit.
I don't recommend trying to learn yoga on your own if you've never done any type of yoga before. There is no replacement for having a teacher be able to show you poses up close and from different angles, and to come around and correct your poses as your practice. It is very possible to injure yourself by doing poses incorrectly, especially repeatedly (I'm assuming you won't do yoga just once and then quit). That being said, Rodney Yee's Power Yoga DVDs are good for practicing at home.
As far as doing both on the same day; if you have no injuries that prevent you from being active, you can do as much yoga as you want in one day. I know people who do 1-2 hours of yoga every day, all the way up to people who practice four or five hours per day.