ooh, when are you visiting?
Roll for the Soul is a nice little vegetarian cafe in the centre, provided you don't mind a bike tire hanging over your head while you eat. Pepe Nero in Broadmead has a separate vegan menu, but the only time i've eaten there they took ages to make our food, and i didn't find it that impressive. there's also a kiosk in Castle Park that makes veg middle eastern food, which my father-in-law recommends. and St Nick's Market has tons of good food stalls, though it gets crowded in the early afternoon & doesn't have much seating.
La Grotta (near Castle Park), Riverstation (harbourside), and Kathmandu (near Colston Hall) all sound nice for the evening, but i haven't been to any of them yet.
there's also a few burrito places & a fair number of (south/southeast/east) Asian places. the local vegan group has a pretty comprehensive guide to restaurants with good options: http://veganbristol.com/centre/