I agree that Avery is best, hands down. I used to work at a computer crafting magazine, and we did lots of projects with transfer paper. Avery worked best for us.
If you can get your hands on a heat press, that will be a huge help. If that's not possible, pay careful attention to pressing hard with the iron and getting each spot for about the same amount of time. As with anything, read the directions carefully before you start. Some transfer papers are hot peel and others are cold peel. (I can't remember which the Avery was.) If you're supposed to peel the backing while it's still hot, peel it immediately when you take off the iron. If you use a heat press and have a lot of small things to peel all at once, get someone to help you so that you have another set of hands. If you're supposed to wait until it's cool, then wait plenty of time before you peel it off.
When washing your project, you should take it out of the washing machine as soon as it's done. Don't let it sit around wet. That will cause problems. Unfortunately, that also means that in order to keep your shirt looking nice, you shouldn't wear it if you'll get rained on or splashed on by a ride. That's hard to avoid while at WDW, so it's possible that your shirt may not last past your trip.
