If you have children, there is Finkley Down Farm in Andover. Its a working farm where kids can play and feed the animals.
There is a Hawk Conservancy near by in Andover where they keep some of the Owls from Harry Potter films and you can observe falconry and also take part.
The Cathedral is lovely and if you walk there from the town ( near the woolworths shop entrance under a stone archway ) there is a fab Belgium Chocolate shop on the left just before the arch where you can but a huge frothy hot chocolate or get yourselves some huge chocolate truffles
There is usually always something going on at Salisbury City Hall Theatre, music or a play.
If you like clubs and pubs There is a nightclub called the The Chapel and Vestry and a Russian Vodka bar called Molokos.
The Odeon Cinema above Marks and Spencer's is worth a visit. Its set in a Old Gothic Theatre and has huge chandeliers and gothic candelabras everywhere. My DH loves it there.
If you go to the tourist information building, I believe you can join a free walk of the Town where they will take you around and show you certain buildings from the old market days. I think Tudor times - but I could be wrong.