I've taken lessons for the past 16 years. Hand technique is very important, because playing with bad technique can lead to serious injury. You don't want your hands to be flat on the keyboard. Instead, have your fingers "curved" a little. Also, don't put stress on any part of the hand, including the wrist. You want the wrist to be loose and relaxed, but to be able have some power as well.
For finger strength, I'd check out finger excercises by Czerny (can't remember if that's how you spell his name) or Hannon. The excercises they wrote are considered standards for beginning pianists. I'd recommend you at least get someone who is an experienced pianist to show you proper hand position. I'd practice finger excercises for at least 15 minutes a day (I did that for about 6 or 7 years) to build up strength in your hands. Also, I'd HIGHLY recommend learning how to read music, if you can't already.