Los Alcazares has a great beach with good bars and restaurants. It's a long beach but if you get a taxi to the "525 Hotel" (North End of Los Alcazeres), there is a good little strip there which leads down to the beach area. You'll find Indian, Chinese etc., along this strip too.
If you drive towards Los Alcazeres from Mar Menor go under the A7 motorway and straight down that road you will find a "Super Valu" shop which has everything. At the end of this road you wll meet the main street of Los Alcazeres where there are good Pizza and Spanish restaurants. One in particular is Ramon's which has a great Tapas Bar to the front with full Restaurant behind.
The Mombasa is now open at the new entrance at Mar Menor and has a great Tapas/Bistro bar, 2 Restaurants, a Pub and a Night Club.
If your going to Cartegena drive to the port area where there is plenty of car parks. This will lead you the shopping areas and streets of the town which is well worth a dy out!
Hope this helps!