I like FBinNY's alternate solution, offered I think partly in jest but absolutely correct in that many bicyclists will stop and offer help to someone on the side of a trail with a mechanical problem.

It is often possible to patch a flat tube without removing the wheel, but it is generally more difficult and annoying and one would only use that technique where removing the wheel is more trouble than this method (for example, the rear wheel on a European city bike with a full chaincase, fender and rack mounts, etc).

Loosen or take the axle nuts off, as well as anything else that may be on the axle that would keep you from removing the wheel (the wheel retention clip mentioned in earlier posts, fender stays, etc.). Make sure to remember the order in which everything goes back together (taking pictures with a digital camera can be very helpful).

Good luck! I think learning to replace an inner tube is a good skill for any bicyclist to acquire, even if they don't consider themselves mechanically inclined. Once you've done it a couple of times, it will become second nature, and depending on how often you get a flat tire it will save you a lot of money over taking the bike to a shop every time.