The Pannier Market in the centre of the town is a traditional market with local produce, art and crafts, brick-a-brac, etc. It's all under cover and there are two cafés there if you need a coffee break. Make sure you check out the famous Roly's Fudge Pantry!
Just down the road from the Pannier Market is Green Lanes, which is a decent-sized shopping centre.
There are loads of little local shops to discover, many of them off the main shopping streets. Try Gammon Walk, Holland Walk and Bear Street.
If you're looking for out of town shopping, you only really have the big retail shops like Curry's, PC World, etc around Barnstaple. For gift and clothes shopping, take a trip to Atlantic Village. It's a shopping park the other side of Bideford and about 20 minutes drive from Barnstaple.