I'm afraid I don't know Torrevieja itself but I reckon that you could
have a pleasant week using it as a base - although you would need a hire
car.

My recommendations would be Alicante - Castle, Museum of Contemporary art
Elche - palm tree gardens also rich early history
Murcia city is also worth a visit and Valencia's science museum is on my
list of things to do one of these days.
Calpe - climb the Penon de Ifach, eat sea food on the quay, take the
boat tour in the glass bottomed boat and you could probably visit  
Guadalest on the same trip.
Monteforte de Cid is somewhere I'd like to go only having seen it on TV
travel programs

I'm assured that there is plenty of walking in the immediate vicinity of
Torrevieja - there's a disused railway line that provides miles of
suitable terrain -
but by going a bit further afield you would be certain to find lovely
walking - in your search for guidebooks look out for the Costa Blanca  
volume in the Sunflower  Landscapes series.
One of my current favourites for walking the completely deserted beaches
of Lorca (another town with a museum) and the area north of Cabo Cope
are only about an hour away via the toll motorways.
This area is due to be developed but at the moment there's no sign on
the ground of this fate.

Sue James