There is taxi-van service that can be used from Split and Makarska.
There is a ferry service from Ancona, Italy. Take a ferry to the Port of Split and than use the bus to get to Omiš.
There are local boats that run several times a day from Islands of Brac, Hvar, Korčula and Vis. Take them to Split and then use the bus or taxi service to get to Omiš.
You can get around the city by car, local taxi and bus service, or tourist trolley. You can also rent a bike or moped from tourist agencies. However most people find it sufficiently comfortable to get around on foot, because Omiš is relatively small city.