From Sibenik to Split there are plenty of bus connections year around, the travel time by bus is about 1.5 hours, majority of buses drive along the picturesque coastal road, to get the best view you should get a seat at the right side of the bus.
The bus departures from the bus station in Sibenik, and in stops in Split also at the main bus station. If you plan also to travel return from Split to Sibenik, you can save money if you buy a return ticket with the SAME bus company. Timetable for returning buses you can see here.
If you are intending it to travel to the Split airport you should take a bus from Sibenik to Trogir and the take local bus nr. 37 to the airport.