I live in between the two so I reckon I might be able to answer this one!
Christchurch Mansion is definitely not to be missed. It's in Christchurch Park which is more or less right in Ipswich town centre. (i.e. it's only a few minutes walk away from the Tower Ramparts Bus station):
http://www.cimuseums.org.uk/Christchurch-Mansion
Ipswich Museum is also centrally located:
http://www.cimuseums.org.uk/Ipswich-Museum
It's probably not the best time of the year to visit Ipswich Waterfront but you'll get to see some ruddy expensive yachts and cruisers and you'll also find plenty of places to eat and drink:
http://www.tripadvisor.co.uk/Attraction_Review-g190725-d1747745-Reviews-Ipswich_Waterfront-Ipswich_Suffolk_East_Anglia_England.html
Moyse's Hall Museum is right in the centre of Bury St Edmunds and definitely well worth a look:
http://www.suffolkmuseums.org/suffolk1/pages/moyses.htm
You'll end up knowing even more than Stuey does about old clocks ;-)
http://www.eadt.co.uk/news/more_gershom_parkington_clocks_than_ever_to_be_on_display_at_moyse_s_hall_museum_in_bury_st_edmunds_1_3974359
One of the best places to eat and drink in Bury St Edmunds is the Old Cannon:
http://www.oldcannonbrewery.co.uk/
(They've got their own brewery, with some of the brewing equipment actually inside the bar!)
Lavenham is one of the most attractive villages in England. There's not much to do there (other than admire its beauty and spend money in one of the many expensive tea rooms) but, if the weather's not too bad, it might still be worth a look:
http://www.discoverlavenham.co.uk/
Further afield, Norwich is packed with attractions:
http://www.visitnorwich.co.uk/
It's easy to get to by train (so you could leave the car at your base and relax more). A day's unlimited rail travel throughout East Anglia (after 0845) costs £18.10 (with a third off if you've got a Railcard) and takes you to places like the North Norfolk Coast or Cambridge:
http://www.abelliogreateranglia.co.uk/tickets-fares/discounts/rangers-rovers/anglia-plus
Cambridge is also easily accessible by road (but use the Park & Ride facilities - don't attempt to drive or park in the city centre!):
http://www.visitcambridge.org/
Tourist info for Bury St Edmunds:
http://www.visit-burystedmunds.co.uk/
and for Ipswich
http://www.allaboutipswich.com/