Lisa,
You do not have to turn the bike upside down. I have seen many people do this and it only damages your bike! I am a mechanical dunce and I haven't had to change a flat on the road in 6 years. But, I have practiced the rear wheel thing recently. Before you take the wheel off, put your chain ring in the biggest ring. Then put the cogs on the one closest to the outside. When you put the wheel back on, you can easily line the chain up on the right cog this way. Then just pull back the derailleur and it falls right in. It's hard to explain in writing, but maybe you could have someone at a shop show you.
This was scarier to me than the actual tire changing part, so I have tried to make sure that i can remember it all....