You cannot easily get between these countries on public transport nor with a rental car. The best option is to fly; see www.skyscanner.com.
If you take trains (which do not run as far south to Dubrovnik), then plan to backtrack from Split to Zagreb, then Belgrad and south. Expect at least a border check in the middle of the night at the Bulgaria/Turkey border, if heading to Istanbul by train. The overnight train from Belgrad to Thessaloniki doesn't cross the border until after 6 a.m.
How to Look Up Train Schedules and Routes Online gives you the DB train schedule link and tips for using it. See also http://www.ricksteves.com/travel-tips/transportation/trains/croatia-rail-passes and http://www.ricksteves.com/travel-tips/transportation/trains/greece-rail-passes.
You can fly between Istanbul or Izmir and Athens, or take ferries from a few coastal towns to the nearest Greek island, which may be far from your other Greek destinations, requiring more ferries or a Greek flight.
Guidebooks like Lonely Planet give a good summary of public transport options for the areas they cover, as do Rick's books (which don't cover Belgrade or some large sections of Greece).