I would definitely go to los Alcazares - the beach is lovely (the sea is very warm and very shallow) and you can promenade for miles. In the summer there are lots of bars and cafes on the beach for refreshments. Market day in los Alcazares is Tuesday, but wednesday is market day in Santiago de la Ribera (on the road past the airport) which is much bigger. My mum said in Los Alcazares square on a friday night there is live music and entertainment right through the summer.
I would also reccommend Mazarron for a day trip - slighly more commercial than los Alcazares with more brits (los Alcazares is very Spanish) but still very nice.
The best shopping around is at Mediterranean park - on the road to Cartegena. There is a shopping mall plus a huge Carre four. There are a number of large supermarkets in los Alcazares which will be nearer to Polaris world. There's a big Mercadonna and Best Value. Another mall can be found at Dos Mares - one of the few places open on a Sunday which also has a large supermarket (Eroski's) attached to it.
I would reccommend a trip to la bamba pizza restaurant in los Alcazares (under the arches) for a very cheap and tasy meal - the garlic bread is the best I have ever had. We normally eat for around 20 euros for 4 adults - the pizza's are so big you can easiely share and the garlic bread is a garlic bread pizza - salads are fantastic too.
Hope you have fun - I'm going back the second week of September with our 2 yr old.