You'll love Alicante.
I'm so envious that you get to be there for that long.
I studied at the University there for a month.
If you are in the city, a bus will take you to the university for a very cheap price. The campus is actually in San Vicente, just to the north. The ride will only take about 10 minutes.
Playa San Juan is a wonderful beach for during the day and at night there are literally tons of people in the clubs near the beach. If you feel lonely for home, there's even a McDonald's right there on the beach.
THere are many small towns near Alicante that are accessible by train.
Also, try taking a day trip to Isla de Tabarca, a small historic island a few miles off the coast. They have boats leaving the port in Alicante many times a day.
Also, there are tons of places near the port in Alicante where you can hang out, eat and meet friends.
Good luck and let us know how it goes.