From what I have read, there are no trains between Dubrovnik and Split. I’m having trouble finding a bus between these two cities.
Flixbus seems to to the trip.
The buses are pretty much hourly. There should be a shuttle bus from airport to bus station (check DBV airport website) , where you’d change to intercity bus (google “Dubrovnik bus station” and that should bring up the schedules.
Flixbus should have you covered. They have a nice app too. Alternatively, you might check Croatia Airlines for a faster, though undoubtedly more expensive alternative.
I don't believe there is a direct flight--it's roughly four hours on the bus. If you don't want to do that, there are catamarans (which take longer, cost more, but schedules are here in case you are returning via the sea: https://www.ferrycroatia.com/en/dubrovnik-split/.)
Airport shuttle to bus station:
https://platanus.hr/shuttle-bus/
Bus schedules:
https://autobusni-kolodvor-dubrovnik.com/en