Here's the official tourism site for Colchester:
http://www.visitcolchester.com/
Colchester's 'star' attraction is probably the Castle Museum. (Some of the smaller museums are also quite interesting). For the best shops, get away from the dreadfully dull High Street by cutting through Red Lion Walk and continuing until you find the more interesting areas which play host to many independent traders.
As Mushroom25 has indicated, Colchester is a good base from which to explore Constable Country. Other places to visit in the area include Coggeshall (an historic small town with several antique shops) and Ipswich (the county town of Suffolk with lots to explore - don't miss Christchurch Mansion).
If you love fine food, a visit to The Food Company is a 'must'. This store is located just outside Colchester, at Mark Tey. It's basically a luxury grocers - think of it as a modern version of Fortnum & Masons - but it's also got a good restaurant. People travel from all over East Anglia to visit the store. The website explains it far better than I can:
http://www.thefoodcompany.co.uk
Chris