Public Transport network in Sarajevo is pretty well organized one. With a combination of Tram, BUS, Trolley Bus and Mini Bus lines almost every single point of the city is reachable, so all you need is to know hot to reach there.
In the central part of the town and for whole flat area of the Sarajevo city, the best is to use first trams, then trolleys and buses. For far up the hill areas, and some special destinations mini bus would be the best choice.
One of the ways we suggest and consider the best is to download free Sarajevo Public Transport Map and suite yourself. Everything else is up to you.