How do I make sure my bike's the proper size?
While in the saddle, your feet should not touch the ground and your leg should be almost completely extended at the bottom of the pedal stroke.  You ought to be able to straddle the top tube fairly easily with your feet flat on the ground.

Sheldon Brown's article on starting on a bicycle may help with getting accustomed to having the seat higher than you are used to.  Do visit his site to see the video, but here are the basics:


  
  Stand astride the frame, both feet on the ground.  Do not try to sit on the saddle while the bike is stopped, this is not usually possible if your saddle is properly adjusted.
  Rotate the pedals so that the pedal for your less skillful foot is 45 degrees forward of straight up.
  Put your foot on the high pedal, then press down hard, and mount the bike as you begin to move forward.