Imposing this type of architectural structure on the rose bush will help prevent powdery mildew in the Spring via good air circulation within the bush. Additionally the overall shape of the bush will be pleasing to the eye when blooms finally appear. Plan to remove all weak or twiggy growth on main canes that is and not capable of sustaining a reasonable thickness of stem. Hybrid tea blooms need a stem about pencil thickness or greater for support. Remove them from as close to the main root cane below the bud union as possible.
Suckers can only appear on varieties that have been budded such as hybrid teas.
The sucker represents the growing habits of the understock used for budding, usually Dr. Huey. If allowed to grow, the sucker will eventually take over the whole bush and suppress the variety budded onto it. Remove old canes that appear woody by sawing them off as close to the bud union as possible. Make this cut clean and smooth. Should any stubs from bad cuts made last year remain, saw them off cleanly.