I went in to the bike shop and had them demonstrate how to change a tube for me. Would you recommend that I practice taking out the tube and putting it back on at home to make sure that I know how to do it? I would have done it already if I wasn't afraid of messing something up.

I agree. I just started biking this past March and was very hesitant to screw anything up on the bike by playing around. But I've slowly learned that it is really, really hard to drastically mess up anything on a bike. Since learning how to do something simple like changing the tube, I have changed out a drive train and shifters on one bike and built another completely from scratch. It just takes getting over that fear of hurting your bike and realizing that you can't really damage a bike beyond repair just by tinkering with it.

You're never going to learn anything about your bike if you don't get your hands a little dirty! Plus, when it comes to changing a tube, really the worst that could happen is that you might pinch a hole in the tube and have to get a new one for like $5. A cheap mistake to make and worth the learning experience.

I went in to the bike shop and had them demonstrate how to change a tube for me. Would you recommend that I practice taking out the tube and putting it back on at home to make sure that I know how to do it? I would have done it already if I wasn't afraid of messing something up.

Changing a tube may seem easy and basic, but it does take practice. Just watching someone else do it is not enough experience. Practice this at home rather than out on the road where you might be left stranded if problems occur.

If you ride, you're absolutely going to have to do this many times. I've had six or eight flats already this summer, and I'm not putting in that many miles. I once had six in my 12-mile commute to work, when they were clearing land for a new subdivision and billions of goatheads blew across the road.
As for being afraid to screw something up...there's not that much you CAN screw up in a tube change. All I can think of is that you might pinch the tube, and that's easy to avoid. Even if you do screw up, it's a lot easier to deal with at home, where you have tools and a place to sit down and no hurry, than out on the road where resources are limited.
A few tips:
Before you remove the wheel for a rear flat, put the chain on the small chainring and the small cog in back. You don't ride that way, but it gives you slack in the chain, and when you put the wheel back, you'll know where the chain belongs. Then open the quick release, pick up the back of the bike and give the top of the tire a sharp rap with the heel of your free hand. About nine times out of 10, the wheel will just fall off the bike.
After you find the hole in the tube, look and feel inside the TIRE for something poking through. It's common for stuff to stiick in there, and if you don't get it out, you'll get another flat instantly.
Whether you replace the tube with a spare (patch it later; there's no reason to throw away a punctured tube) or patch the damaged tube on the spot, when you re-insert it, blow it up until it holds its shape before you stuff it back in the tire. You're much less likely to pinch it with a tire tool that way than if it's completely flaccid.
You can do this by mouth with presta valves--it doesn't take much pressure. With schrader (car-type) valves, you'll have to use a pump.
After the tire and tube are back on the wheel and reinflated, reinstall the wheel on the bike. This is the awkward part for many beginners, but since you know the chain's going to go back on the small cog (because that's where you left it, remember?), that's easy.
Depending on your brakes and tire size, you may have to open the brakes or unhook a straddle cable to get the tire past them. That's simple, too, and if you can't figure it out, the shop can show you in about two seconds.

here'smy advice, changing a tube is not as easy a you think. thehardest part is getting the tires to go on the rims on top of the tube. the sidewalls of the tires ar so reinforced that its a bi*** to get it in. ive had my hand blistered because i had to put on the tire, inflate then find the tube punctured, remove, put tire back on reinflate, find out tire is broken again yadda yadda yadda. but then that was stupidity on my part when i forgot that i didn't put any clincher tape. anyway experience is better

Ok, well I practiced. I took the tube all the way out, then put it back in and put it together. Wasn't too bad. I feel confident that I can fix it on the road if I have to now. Now I just gotta get a repair stand and tools and learn to do other stuff with my bike.