From north to south we have;
 
Columbretes Islands (Castellón): about 25 miles from the peninsula. It is a set of 4 small archipelagos that people visit to observe the great diversity of both the animal and plant species that it offers. It is a natural park with a unique habitat in the Mediterranean where you can snorkel and relax for a day.
https://www.visitaislascolumbretes.com
 

Tabarca (Alicante): it is the only inhabited island of the Valencian Community, located 11 miles off Alicante and very close to Santa Pola. It is also an archipelago with a length of 1800m where you can find beautiful coves, a very characteristic village and several restaurants where you can spend the day enjoying a purely Mediterranean day.
http://www.alicanteturismo.com/isla-tabarca-alicante/


 
Perdiguera Island (Murcia): It is a volcanic island that is linked to a small peninsula located in the Mar Menor, it is the second largest being 26 hectares in size. Here you can find chiringuitos where you can enjoy the typical Murcian cauldron or fish of the area, a great tourist activity in the summer months. It was established as an island of great tourist activity, which meant deterioration in the vegetation and quality of the area. In 2007 the Law of Coastal Demarcation established the elimination of these establishments and we saw the return of a form of tourism which is more sustainable. 
http://www.allyouneedinmurcia.com/en/naturaleza-mar-montaña/isla-perdiguera/


 
There are also the Balearic Islands, of which we will talk about more carefully in another post.