Grape vines, they take some love but don’t wander.

Scarlet runner is a fun one to grow its a bean with red flowers on it and it gets pretty tall and has nice foliage.

Try Clematis one of my favorite plants

Climbing roses and clematis are non-invasive and have the benefit of beautiful flowers.

Climbing roses come to mind as well as clematis. They both come in a wide variety of shapes and sizes. My favorite rose is “Lady Banks”, It has no thorns small leafs and 2″ double yellow flowers which cover the whole plant every year. It must be pruned to shape as most wandering roses. My favorite clematis is, “Nelly Mosier”. It’s two tone pink flowers are 4-5″ and it blooms in May thru June in my zone 7. These two plants mind their manners and do require prunning every couple of years. They don’t grow out of control and stay in one place.
Check with your county extension service for references and local gardens to tour and nurseries in your area to view the plants you are interested in. Their are so many other’s to choose from. Good Luck