Is there one search engine of scheduled buses in Italy, where I can search timetables of any bus company? Or, if not all of them, at least significant number of companies.
Every city has its own local bus company with timetables and lines:
Take the train could be a good choice to travel across Italy:
Last but not least, consider Google Maps search. It supports public transit in its "Got Directions" pretty well.
Enjoy your trip!
Italy is also covered by Google transit. I don't know how good it is for Italy, but for my home town it is quite good.