This article addresses modern climbers. Many climbing roses are old garden roses. Those climbers are treated under the discussion of old garden roses.
My favorite climbing rose for Houston is Don Juan. It is a fragrant red rose that blooms prolifically, even in our heat. It is relatively disease free and will be covered in blooms by the second year if you remember the key trick to all climbers: train the canes so they grow as near to horizontal as possible.
If you are having trouble with your climbers having blooms only at the tips of the canes the problem probably is that you are letting the canes grow too close to vertical. Bend them over and they will reward you with blooms all along the cane.
Also, don't heavily prune your climbers. You need those long canes to produce the blooms.