Are you left handed or right handed? Are you picking with your left hand or is your left hand the hand that you are placing on the frets? If your left hand is your fretting hand, you want to make sure that your hand is in proper placement on the guitar neck before you start to play. If you're just starting out, a few lessons would probably save you many hours of frustration, plus it will help prevent you from developing bad playing habits. Hand placement is much easier to show someone than to describe in words, but I'll try:

First: Take your hand and make a 'letter C' or a claw (however you want to think of it). Your thumb, (or the bottom of your "claw") should remain curved and be placed on the back of the guitar neck towards the middle of the neck (referring to the middle or the centerline of the guitar neck). You will actually use your thumb as sort of a pivot point when you play. It will feel a bit awkward at first, but its important to keep your thumb toward the middle of the neck and somewhat curved when you play. You don't want to wrap your thumb around the neck when you sit down to play. Some players with really big hands can get away with wrapping their thumb all the way around the neck while they play, but if you don't have big hands, get out of this habit as early as possible.

Second: When you are placing your fingers on the frets, you want them to be curved as well, sort of keeping with the curve that you made with your hand when you made the letter 'C'. Try visualizing your fingers as piano mallets hitting piano strings. When the mallets hit the strings on a piano, they hit them in a downward fashion. If you keep your fingers relatively curved when you play, that will help reduce unwanted intrusion by your idle fingers while you're playing.

If you are still having trouble, many beginning guitar instruction books have a diagram for proper left and right hand placement in the beginning of the book.