FIRST DO THIS:

Every single time that you get a puncture, check the inside of your tire to see if the thorn or whatever caused the puncture is still stuck in there. You won't find it every single time, but you will find something often enough to make it worth the effort. Also, if you leave the thorn or whatever hidden in your tire, you can bet it's going to puncture your new inner tube on your next ride. A good trick to help you search for the thorn is to get into the habit of always aligning your tire label with the valve stem. That way, after you find the hole in the inner tube, you will be know where on your tire to search for the thorn.

NEXT:

There is no puncture proof, there is only puncture resistance. I think that if a product is properly designed, it shouldn't require you to buy an accessory to make it work. Bicycle tires, even tires from the same manufacturer, come in a a variety of puncture resistance levels. The puncture resistance belts really do work, but at the expense of ride quality. The most puncture resistant tires feel to me like riding on a solid wheel. If you think you're getting too many punctures, look for tires with better puncture resistance. If you aren't getting very many punctures and are looking for a a little better performance or a smoother ride, you might want to take a chance on tires that aren't quite as stiff.

SLIME AND MR. TUFFYS:

Slime is a sealent that you squirt into your inner tubes. It works great so long as you don't use too high air pressures. Sooner or later, however, you are going to have to change your inner tube and it's messy.

Mr. Tuffy is a brand name for tire liners. They go between your tire and inner tube. To install them, lay your tire on the floor and put just enough air in your inner tube to give it shape. Stick the inner tube in your tire and stick the Mr. Tuffy between the tire and tube. The bad thing about Mr. Tuffy's is the end can wear on your inner tube and eventually cause it's own flat. It'll take awhile before it does that however so, if you're getting a lot of thorn flats, it might be worth it to you.

If you ride with a group regularly, my advice is to buy the exact same tire the majority of them use. That way, if you do puncture, you won't have to endure a lecture about what a stupid tire choice you made while everyone is waiting for you to fix your puncture.