Depending upon your zone, there are many climbers you can mix and match- climbing roses with Clematis, native honeysuckle (Lonicera semperviren), Crossvine (Bigonia capreolata) which is very fast and can be quite aggressive but it is native for the hummingbirds. I do not know how large your arbor is but you could easily add in different flowering and green vines. Virgin's bower (clematis virginiana) and even Virginia creeper are options to add in
Many climbing perennials are very invasive so be careful of your choice.I would opt for the lesser and combine different clematis.
We're in zone 5 and Sweet Autumn Clematis is my covering vine of choice. After it acclimates the first year, it will really take off (typically 15 feet or more) and blooms last for 5 weeks. Since it blooms later in the season, I'd recommend mixing it (as others have said) with maybe some morning glories, moonflowers or climbing sweet peas. Good luck!
I would want to plant something that could be contained to area you want it in and wouldn't attract bees. Consult with a nursery.
scarlet runner beans are fast if that matters...only an annual though...
Mandeville are summer beauties-climb and come in red, white, pink, and a striped pink and white. Will not winter over so easy to get rid of in the fall. Sometimes some of the vines tend to take over but these never disappoint.
Try trailing potato vine-purple in color with a bright green one-annual only so easy to get rid of in the fall.
Hops grow fast and give a lot of coverage in the summer.
I agree with the folks above - research what climbers grow in your zone, and consider whether or not you want them scented and/or with flowers. What does the vine look like when there is nothing blooming?? Some of them still look nice, others not so much! We're planning on ivy geranium for ours - it's very easy to take care of, comes in all kinds of different colors, is in full bloom for quite a long time of the year, and still looks decent when there are no flowers. Good luck! :-p