HERE ARE JUST A FEW of the easy, dependable vines that can be grown on arbors, trellis, fences, posts, or anything else vertical. Most are generally available, at least in locally-owned garden centers (they can ORDER the vines from their normal wholesale channels; a good garden center manager will do this for you. He or she isn't willing to do this simple act of customer service, ask them where they want you to shop that WILL).
And by the way, it is PERFECTLY okay to plant two or more vines on the same support. Really.
Carolina jessamine ("yellow jasmine")
Clematis (including fall-blooming "sweet autumn" clematis and the evergreen Armand's clematis)
Climbing roses: Lady Banks (thornless, once bloomer), Climbing Old Blush (pink), New Dawn (large pale pink), Zephirin Drouhin (pink, nearly thornless)
Coral or Trumpet honeysuckle (native, not invasive, great for hummingbirds)
Cross vine (native, mostly spring blooming; look for the more prolific Tangerine Beauty)
Cypress vine (fast annual with hot red flowers)
English ivy (best in shade, can be invasive if not kept in bounds)
Fatshedera (cross between the Fatsia shrub and English ivy, a semi-vine that also comes in attractive variegated forms)
Gourds (incredibly fast annual, many different kinds, great for kids gardens)
Hyacinth bean (annual with burgundy tone stems, lavender flowers and burgundy beans in late summer)
Japanese honeysuckle (very fast, great for hummingbirds. Considered too invasive by most gardeners, but...)
Moonflower (fast annual with evening flowers)
Trumpet vine or trumpet creeper (fast native with great hummingbird flowers; considered invasive by some gardeners)
Wisteria (consider using the native, which is not quite as invasive as the import from Asia)
Blackeyed Susan vine (fast annual)
Confederate jasmine (evergreen with fragrant flowers)
Passion vine or "maypop" (natrie, with edible fruit. Considered invasive by many gardeners)
Silver lace vine (considered invasive by some gardeners)
Virginia creeper (native, great fall colors, considered invasive by some gardeners. Looks like poison ivy except it has FIVE leaflets per leaf)
THERE ARE OTHERS, of course. These are just some of my favorites that do well and are easy to grow.