If you are looking for a place to have as a home base while you are there, the villa I live at now rents flats to tourists. It is in Viareggio, which is just over the mountain from Lucca (about a 20 minute drive). From the villa you can see the Mediterranean Sea. Viareggio is a very fun town for younger people. I attend Italian classes there as well. They have a great school with classes available for all levels, it is 4 hours a day, M-F and goes in 2 week sessions. They also have free excursions to areas nearby and social activities at night. I can't remember what I paid, but it was reasonable.
Also with Viaregio being right on the sea, you can go to the beach, etc.
Viareggio is great because you can very easily get to Pisa and Lucca (each about 20 min away) Florence is 45 min to 1 hour, You can get to Rome or Naples via the high speed train in 2 and 3 hours. Siena is just south of Florence, though you need a car to go there. Just up the coast is Forte Di Marmi, which is very popular with young Italians for discos, etc. Cinqueterra and Portofino are north of there.
If you are interested in more information about the villas, let me know. And I am sure a discount could be arranged if you were to stay for that long.
Good luck and have fun whatever you do!