Hello Jen,
What a great way to learn Spanish! I suppose whether you choose Valencia or Alicante will have a large bearing on your tolerance for crowds. Alicante is a town whereas Valenica is the capital city of the Comunidad Valenciana (of which Alicante is a part).
If you meet and make great Spanish friends in Madrid, chances are you will see them again in Alicante as it is one of the summer hotspots for Madrileños...especially La Playa de San Juan. I have no doubt that there is a nightlife in Alicante. Like many of the coastal towns, it triples (at least) in population during the summer months. The beaches are very very crowded.
Valencia, as a capital city, has a lot more to offer than just beaches and is more centrally located in case you wanted to catch a train or bus to visit another town or city during your stay (I highly recommend Peñíscola and Vall d'Uxó)....if you are looking to buy leather goods (coats, clothes...) there is a inland Valencian town called Llombay (or Llombai in Valenciano) that is chock-a-block full of shops specializing in leather goods.
The largest beach in Valencia is Malavarossa and is accessible either walking or by bus. I am not sure if the metro goes out near there or not. There is also a wonderful nightlife in Valencia, to which I can personally attest....especially in areas such as Canovas, Plaza de Xuquer and Barrio de Carmen.
Good luck with your decision!
-R
