My favorite annual climbers are morning glories. They're easy to grow, fast blooming especially if you start them early, and will bloom until frost. There's a wide array of colors, and they usually self seed.
Colleen

If you want perennials, then of course clematis are great. They're easy to take care of, despite their reputation as fussy or fragile. I've moved mine repeatedly and don't bother with the whole "keep roots in the shade" business; they do fine and I haven't killed one yet.

The annual that I'm eager to try this year is cardinal climber. Even before they bloom, the leaves are so pretty - they are palmate and have an asian effect. I made the mistake once of growing cypress vine (ipomoea quamoclit), which I thought was cardinal climber (ipomoea sloteri).. I was so disappointed!