Vienna to Croatia

Easiest would be to fly, Croatian Airlines offer direct flights from Vienna to both Split & Dubrovnik. Although ticket prices aren't very affordable in this case, so might make sense to fly with change and save some money, for instance Serbian JAT airlines offer more or less good fares for flights Vienna-Belgrade-Split/Dubrovnik
If to speak about going by train, then there is not much of a choice - train from Vienna to Zagreb (discounted tickets start from 29 euro) and then another train to Split (ticket is app. 25 euro + need to add price of potential sleeper reservation). Dubrovnik doesn't have rail connection anymore, so you'll either have to take a bus from Split to Dubrnovik or bus from Zagreb to Dubrovnik
Another interesting possibility, but only for those who have time, is to take train from Vienna to Rijeka (at least one change will be required), while from Rijeka you can catch a ferry to Split and Dubrovnik, journey by sea is really nice, though takes longer than going by land