Do you need edge banding or something more substantial?
A pretty simple way to make a contrasting wood ring for the outside edge of a round table is to make an octagon of the wood you'd like to use and then simply cut an inner circle to fit the table. This is probably easiest done on the band saw, just cut you an entrance, you'll have to cut this ring in half for mounting it to the table anyway, and use a circle cutting jig. Then once you have that inner circle cut out, make another cut 180 degrees away from your entrance cut, fit it to the table, glue it and clamp it and come back later with a router and a straight cutting bit attached to a circle cutting jig and cut the outside circle. I'd do the outside circle last simply because the octagon makes it easier to clamp while the glue dries. I've done this several times and it's really not very difficult. I use biscuits to position the ring, but you could use breadboard/tongue-in-groove bits too.
I hope I explained this okay, re-reading it makes it sound sorta like a pain, but really, you can do this in just a few hours and probably not get overly annoyed.