The bus is easy, convenient and least in expense and you will tuck your luggage in the bus luggage compartment. Here is how it works:
First, the Split airport is relatively small (single runway) and easy to navigate.
https://www.google.com/maps/place/Split+Airport/@43.5367083,16.2975767,688m/data=!3m1!1e3!4m5!3m4!1s0x133542620883b3e9:0x7b7b3553c3140f87!8m2!3d43.5367044!4d16.2990251
Exit the terminal and keep an eye to the left side of the parking lot seeking the typical tourist size passenger buses. There may be several buses, just walk up and ask which bus is direct service heading for the Split harbor. You will be pointed to the right bus.
The compartments under the bus will be open and you will place your luggage in them. Overhead racks are tight so recommend placing everything under the bus. Do not worry about theft! This is not that type of environment and the driver is also keeping an eye on the compartments.
I believe the cost is 40 kuna each and payable only in kuna. Be sure and exchange currency for kuna. Board the bus and the driver will come and collect the fee prior to departure. Relax for the ride is about 35 minutes depending on traffic.
The arrival station at the port is just as simple. Exit the bus, grab your luggage and head for your accommodations. Split is easy walking distance and your can cross the entire palace from east to west in less than 30 minutes. Most folks choose to walk to their destinations.
Croatians are wonderful kind folks and we greatly enjoyed our interactions with their culture.
Enjoy!