Well, first off, if both your feet can touch the ground while you're sitting on the seat (saddle), you're probably going to hurt something by pedaling that bike, most likely your knees, but you might also make that hip worse. When you come to a stop on a bicycle, you should get out of the saddle. Proper seat height for a bike is such that when the pedal is at it's lowest point, your knee will be only slightly bent. When you pedal, you should be able to do this with each leg without rocking back and forth in the seat.
As for size, you're probably going to need a youth bike of some sort, as I'm guessing custom is out of your price range. For a wild guess, I'd say around a 46-48cm frame. Best bang for your buck may be to try to find a used bike in good condition.
With regards to shifting, a common mistake I see people make is to wait too long to shift down. You have to be pedaling to shift, and it takes a certain percentage of a revolution of your crank to make the shift happen. The slower you're pedaling, the longer that shift will take, and if you're trying to muscle up a hill at the time, you may run out of muscle to keep going in the higher gear before the shift completes. This is also when the chain and derailleurs are under a lot of tension and stress, and the chance for something to go wrong is at its highest.
However, if it's relatively flat in the area where you'll be cycling, a single-speed bike may fit the bill, and is certainly much simpler with fewer mechanisms to maintain and adjust.
