The airport is relatively small but serves a fair number of domestic and European cities. If you're not able to fly to your destination from Split, there are daily flights to Zagreb, which is a bit more connected to the outside world. The airport can be reached by hopping on bus number 37 (bound for Trogir) from Sukoišanska. The bus runs every 20 minutes on weekdays and every 30 minutes on weekends. If you happen to have the distinction of being a Croatia Airlines passenger, you can take the airport bus that waits at the beginning of the Riva, an hour and a half before Croatia Airlines flights. If you didn't manage to blow all your kunas on coffees on the Riva, you can exhange them in the offices run by Splitska banka or at the post office, both of which are located in the main hall of the airport.
And if you did manage to blow all your kunas on shopping, enquire at the information desk where you can confirm your tax-free form, as the office is not visually marked.