700C tires were first introduced to fit on bikes with sew-up tubular tires, to provide a more durable training tire. So a 700C rim is the same diameter as the tubular rims- You should get set up with 700C cllinchers- (They won't fit on the sew-up rims- the rims or the whole wheels need to be changed.

you probably want a decent road bike set with a rim about 3/4 of an inch wide (19mm) or else an inch wide (25mm) if you want a little bigger tire. The small rim will take a 28mm or less wide tire, even some 32mm x 700C- they generally aren;t labled real standard-like between different tire makers- so sometimes it's best to see them before buying off the internet