I see a number of questions on this forum about public transportation to/from the airport in Sarajevo. On my last trip I tried it to and from and wanted to report back.
First of all, I used these instructions for the Trolley bus: http://www.visitsarajevo.biz/travel/airport/. Essentially, you have to walk in through the neighborhood of Novo Sarajevo and catch a trolley bus into the center. The good news is that the instructions work, are easy enough to follow, and it's very cheap: 1.6KM, which will be much less expensive than a taxi. You just buy a trolleybus ticket from the kiosk at the bus stop - it helps to have change. Of course, be sure to punch your ticket when you board to validate it. (The punch is not electronically stamped like the tram tickets.)
The bad news is that this route is a hassle and takes a long time: it's a decent 20 minute walk at least to the bus stop, at times without sidewalks and along heavy traffic, not something I'd recommend with much luggage or in the midday heat. All told, it took me about an hour to get from the airport into town.
There is a quicker route than the one described above, which you can find via a good map like OpenStreet; this cuts more directly through the neighborhood. However, the roads are not direct and it's difficult to describe in words if you don't know where you're going.
In short, I think most tourists, particularly first-time visitors, are better off just paying more for a taxi or (better yet) arranging a shuttle through their hotel. But if you are committed to public transport, it is doable.