You can drive in Indonesia on an international driver's license. Motorbikes and cars are widely available for rental (if not provided by your hosting organization or offered to you by your host family) and one liter of petrol costs only 4,500 IDR (half an American dollar), eventually you will want to give it a try (don't forget the left-side traffic!). If you are not an experienced rider though, Indonesia is not the place to learn and in most cases it is better leave it to the locals to drive you around.
In Indonesia, the traffic moves on the left hand side and cars overtake on the right. The rules and regulations differ from other countries and in many places they are not taken seriously.
Many of the streets in Indonesia are quite narrow and driving in these tiny roads can become difficult especially if you are accustomed to wide open roads. A lot of Indonesians park their cars on the street thereby restricting road space to drive on. You have to be extremely careful of motorcyclists, pedestrians, buses etc while driving as they may cut you from any side to pass.
The best value option for long distance transport is the train lines. These are comfortable, reliable and efficient. They also have the bonus of avoiding traffic (which lengthens bus journeys by about a couple of hours), meaning that they stick to their timetable. There are 3 different classes of travel available so the price is also flexible to your budget.
Public buses are very cheap in Indonesia but are typically quite slow. They will make countless stops to pick up other passengers and are obviously affected by the prolific traffic jams in the country. They can often be extremely cramped and rather bumpy. However, they can be quite an enjoyable experience for shorter trips as your companions will doubtless include not only people but also assorted livestock accompanied by be locals who rarely see tourists and will be fascinated by you. For long distance journeys not served by train, it is probably worth paying the extra money to get an overnight luxury bus.
If your budget will allow it, internal flights are another option for getting around the country. Indonesia is served by a fairly comprehensive internal flight network flown by a number of local companies. These are more expensive still, but are the quickest way to get around.
If travelling to large islands like Bali, a number of large ferry services operate to other islands in the vicinity. If you go to some of the smaller islands (e.g. The Gili islands off northern Lombok), boats are much smaller and a bit rickety. In good weather they are fine, but in rough weather the ride can be a bit turbulent.
In Bali and Lombok, hiring a car can be a good and relatively cheap way of getting about. Hiring a car with a driver / guide is marginally more expensive, but is perhaps a better idea as it will mean avoiding all the problems associated with driving yourself about.
Selamat jalan and hati-hati!