What is the best way to travel from Brindisi to Salerno ?
It depends on what you mean by "best"?
By car it's a 3.5 hr drive. I've done it from Taranto to Salerno. It's mostly freeway or nice highway.
By train close to 5 hours or more, with one or more train changes (via Taranto or Benevento being the easiest).
By bus I don't know.
For train schedules visit:
www.trenitalia.com