From my understanding a screen protector (a good one like Zagg) will help your screen from getting broken but any fall no matter what you have for protection hits the right way it will break. There are a lot of good suggestions in this thread from the members. When looking for a case look for one that sits a little higher above the screen edge to give protection and also one that has a little substance to it. I would steer clear of the hard plastic "designer cases" which in my opinion offers little to no protection. The best cases are going to be the otter box or similar.
I personally don't use a screen protector (the Bionic uses Gorilla Glass) and I use a TPU case that is a mix of rubber and plastic.
Here is a TPU case for the S3 just to give you an idea. Amazon.com: galaxy s3 tpu case