Indonesia is made up of hundreds of little islands, and public transport adapts to this, so don't think that planes and boats are the only ways of getting around. These are the main means of transport you will find during a trip to Indonesia.
You should think about using the plane as an efficient, inexpensive, means of transport for getting from island to island in Indonesia. There are several airlines, trustworthy and less so, which have flights between the main towns. In any case, those flying between Jakarta and Bali, the most popular destinations, are often safer because the planes are usually bigger and newer.
The second option is a boat. It's a great way of exploring the different islands on the archipelago. The ferry network is well developed with lots of routes. But contrary to what you might believe they are not necessarily more affordable than the plane. So you should compare prices before buying. Of course, you aren't getting the same experience on a boat as on a plane, because as well as seeing the scenery, a boat is a good place to meet the locals.
The most common means of transport for the locals is the bus, because it is very cheap. So it's a great way to get to meet the locals! However, you should be aware that buses are often overloaded and not very safe. And it's not always easy to see where they are going either.
On the smaller islands it's easy and cheap to rent a motorbike or a car. It's great if you want to go off exploring by yourself, at your own speed.
Finally, there is also the train if you are on the islands of Java and Sumatra. Indonesian trains aren't very modern but they do have charm. It's another excellent way to get to know Indonesians and have a chance to talk to them while you travel.
Finally, unless you are taking the plane, it doesn't really matter what means of transport you choose, you'll travel through amazing scenery and meet pleasant people during your trip to Indonesia.