Most climbing roses bloom on old growth, that is, the growth from previous years.

The best time to prune these roses is right after they finish blooming (typically in May). If they are repeat bloomers, then prune them right after their major spring flush.

Proper pruning would be to remove the oldest canes entirely (all the way back to the main trunk), although some secondary pruning can also be done.

Regular maintenance (tying, etc) will be needed through the growing season to keep them on your arbor or trellis.

The University of Georgia has an on-line publication for rose gardening:

Rose Culture for Georgia Gardeners

See also these tips:

Pruning Hybrid Tea Roses and Pruning Roses