Inspect the tire:
It's important to find whatever caused the flat and remove it. If you
don't, the sharp item might still be in the tire where it'll just pop
your new tube. To find it, remove your glove (or use a rag), and run it
around inside the tire in both directions (photo). If something
sharp is still stuck in your tire, it'll snag the glove. Remove the
nasty. If you can't find anything, it's likely it got knocked out during
the disassembly procedure.
Tips
Be sure to check inside the rim, too. The rim strip covering the
sharp spoke holes (and sometimes the spoke nipples), can move, which
allows the tube to be cut. Make sure that the rim strip covers
completely.
As long as the hole in the tire is about 1/4-inch long or less, you'll be okay reusing it.
If the tire has a large gash in it, you'll need to patch it
somehow before reinstallation. Paper money works great. Simply place it
over the hole as you install the tire and tube. It'll reinforce the tire
at the hole and get you home. Replace the tire ASAP.