Well the obvious answer is to hitch - that'll lower your transport costs massively ;) It all depends on whether you're up to trying that. I don't tend to, personally, but on the occasion I have, it's pretty fun.
One thing to note about the public transport on Sicily is that it's nearly a 6 day service - the transport options are massively reduced on Sundays. Check any timetables in advance and check with locals.
Apparently the rail network is cheap compared with other locations in Europe. However, aside from routes between the main cities (Messina, Palermo, and Catania), the trains can be few and far between, so be sure to always check timetables.
Note that sometimes a service listed is actually a bus(!) leaving from outside the station!
Option B is buses - the routes are far more extensive and cheap. Most cities and even small towns have at least one bus linkage to other cities every day. And from anywhere, you can get a bus to Palermo (I'm assuming you're flying into there, which would be useful, but if not, most of this still applies).
There are also some boats to the nearby islands.
To get to Mount Etna, there are daily buses from and to Catania and Nicolosi.