I have a long swath of 6 ft. white fence in my back yard that I'd like to beautify with trellises and nice climbing plants. They get full sun and will be right next to my vegetable garden, so I'd like something flowering to attract bees, but also something that will not get too bushy. Any recommendations?

Any type of Clematis, they are well behaved.

Another idea would be climbing nasturtiums, which would be an annual but may self-sow. I've planted them with sunflowers and it makes quite a splash. The nasties use the sunflowers to climb, and would also use your fence. Both would attract bees, and if you wanted to keep the height down, you could go with the shorter sunflowers.

I also second (or third) the clematis idea.

Thank you all! I'm looking into the clematis idea. . . there are so many to choose from!

There sure are! I found out that some are only two feet tall and some 10-12 depending on the height of your fence you can decide. Also they have big showy flowers and are non invasive!

Check out our thread about, 'Major Wheeler', bigger but also non invasive! Hummingbird magnet.
The thread "Coral Honeysuckle (Lonicera sempervirens 'Major Wheeler')" in Perennials forum

Passiflora aka Passion Vine is a good choice because you get fruit you can eat and they love blazing hot sun.

Bonehead: what a great idea.I raised trailing nasties in the garden and in pots. The container ones didnt do well.I believe the soil gets too warm.
They do self sow in the garden.