In reply to abr1966:
I'd second Bury St. Edmunds, but it's going to be outside of your 20 mile radius. It's a lovely place though. Newmarket is kind of OK-ish. Do not, under any circumstances, go to Haverhill. You'll probably have your bike nicked. Heading North-East-ish kind of direction is Ely, which is also pretty nice. A couple of nice cafes and pubs there.
The ride out to Dunwich/Southwold area is lovely - I've done it a few times.