Bignonia capreolata aka crossvine is absolutely stunning and doesn't spread like Campsis radicans. Take a look:

Here is a link that might be useful: Bignonia capreolata

Yes, that looks perfect. Thanks.
When I looked for descriptions, it said slow to establish. Does that mean a slow season/first year, or that it takes 5 years for any significant coverage? Our backyard is small, and I'd love something better to look at asap.

All perennial vines take time to establish. 3-5 years is typical but some, like climbing hydrangea, can often take much longer.

Native coral honeysuckle (lonicera sempervirens) has become one of our favorites for fences. Improved cultivars are available in (at least) red, orange, and yellow. Fairly quick growing, but stays in its place. Photo shows a 'Major Wheeler' in early bloom started from a 4" pot two years ago on an unshaded east facing fence. It's now covering about 6 ft x 6ft, but will need a couple more years to build that nice looking density and a bit more width across the top. Big flush of blooms in the spring, with minor repeats throughout the summer. Semi-evergreen in mild winters, but cannot hold onto its leaves into low teen temps like crossvine.

Will honeysuckle climb fences? I was told that it needed some sort of trellis.

I just planted major wheeler to grow up the side of our deck, but I could definitely go for a scented variety or two. We have a small and fairly ugly urban backyard, so vertical greenery will be pretty important if I want a lush feel.

Coral honeysuckle is a twining vine that will start climbing on itself once a few vines have been helped to the top of the fence (or whatever). We use 3 to 6 wires on standoffs to get them started. Once a new vine is tied or wrapped to the bottom of a wire, it will usually twine itself to the top.