A lot of tourists rent scooters which are usually Honda Activas, an ungeared scooter which is quite easy to ride. One can also rent geared motorcycles including the popular Royal Enfield, the British motorcycle which is now manufactured in India. This is even bought by tourists from the rental company if they plan to stay longer. However make sure you are careful while riding especially if you plan to ride on the highways. The rental company doesn't provide helmets and it is up to you to look for one if you want one. One can get these scooters or motorcycles from garages or from people standing on the road next to these scooters and one can expect to spend between Rs.150 - Rs. 250 a day on a scooter and a little more if one is looking for a geared motorcycle.
If you aren't comfortable riding in Goa or have a bigger group you can also rent a jeep. These are usually either Mahindras which are similar to Jeep Willys or Maruti Gypys which are the long wheel base version of the Suzuki Samurai. A lot of these jeeps are open without a roof and can really help to enjoy the Goa experience and is a better way to travel around as opposed to a regular sedan. One can expect to pay around Rs. 700 - Rs. 800 a day for these jeeps.
You can also use the local buses to travel to different beaches in Goa. Note that these buses are not owned by Goa municipal authorities and as such the fare can vary. A typical bus ride will cost anywhere from 4-6 rupees. Fares are not collected at the bus doors but rather after you after you entered and the bus has begun to move. It is prudent to ask your fellow passengers about the fare and the destination if you are not familiar with the distance or the place.