Alicante - beautiful cathedral in the centre, and a castle which has fantastic sunsets. The Barrio Santa Cruz area has some beautiful winding streets, flower filled terraces and great views over the city. You can take the bus / tram out to Altea, Vilajoyosa, Calpe or Elche which has its own beautiful cathedral and the largest palm tree park in Europe. There's plenty of great tapas bars and restaurants, vibrant marina area and it's all really walkable too.
Palma - great for the cathedral and for Castillo del Belver. Lovely marina area, and plenty of shops, bars and restaurants. It has a similar vibe to Barcelona, in my opinion, but on a smaller scale. The train to Soller is also worth a day out.
Tarragona - easily reached from Reus airport, around 20 mins in a taxi. Fantastic Roman ruins, lovely coastal walks, plenty to see and do for a few days.
Murcia / Cartagena - fly into Murcia Corvera airport. Murcia is around 30 mins in a taxi, Cartagena around 20 mins. There is finally a public bus service from the airport into Murcia too, but you will need to check timings as it's not as regular. There is also a reasonable shuttle service, that needs to be pre-booked, to both Murcia and Cartagena, run by airportbus.es
Both great cities, both quite compact, so would make for a more relaxed weekend of pottering rather than action packed. For Murcia, you could probably see everything in a day, but there are regular trains / buses to Cartagena, Lorca, Elche, Orihuela and Alicante for a day trip so I would choose to stay there as a base to explore other places. Cartagena has the benefit of being on the coast though, with a Roman amphitheatre, 2 castles ( Concepción and Galeras), marina and a great coastal walk along to Trincabotijas Alta Batteria - an old battle /lookout station which is quite impressive, so it has plenty to keep you busy for a few days too.