There is a nice little bus terminal on the main road next to the beach, between the marina and fort (very close to the marina and Armenian Memorial). This would be a great place to get off if staying in that area. There are also longer distance buses that leave from here to Limassol, Paphos and Protaras if you do not want to hire a car. We ended up getting a great deal on a car rental for a few weeks although did not pick it up until our time in Larnaca was over. I really wouldn't recommend a car while staying in Larnaca as the roads are very narrow in the old city, there are lots of one-way streets making navigation difficult and parking is scarce. It was easy to get around by walking to many of the main attractions in Larnaca. Better to spend your money on watersports, then pick up the car before exploring more of the island in my opinion.
Note: There are buses from directly from Larnaca's Airport to Nicosia, Limissol and Paphos if that is where you are heading to.