"Playa de San Juan" is a very good beach to go to, just 20 min by car or by tram from Alicante city centre. This beach is far less crowded and cleaner. Also there are really good reastaurants just by the sand and not expensive, you can have some of the best paellas and tapas around.
I'm from Alicante myself and this is one of the beaches that I go the most, I find it very relaxing.
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