Don't bother with Newmarket. It's fairly dull and it's main attraction, the National Horse Racing Museum, is closed until the end of March.
Stowmarket has nothing to offer except the Museum of East Anglian Life (which is best visited when there's good weather):
http://www.eastanglianlife.org.uk/
Downham Market and Swaffham aren't worth going out of your way for (unless you want to try to spot the locations for the TV series 'Kingdom', much of which was filmed in Swaffham).
For excellent theatre, arts and sports facilities, together with great shopping and some superb eating and drinking experiences, it's worth the 30 mile drive to Norwich:
http://www.visitnorwich.co.uk/
If the weather's good, it would be worth driving to Southwold, which is the jewel in Suffolk's crown. It's a delightful seaside town, which is well worth exploring:
http://www.exploresouthwold.co.uk/
Chris