Bakeries and bread hold a very special place in the Bosnian psyche. If you open Google maps and search "Pekara Sarajevo" (pekara is the Bosnian word for bakery), you'll be confronted with a sea of red pins, and these are just the ones registered with Google - many aren't. I would say that you haven't really been to Bosnia and Herzegovina if you don't eat at least one thing from a bakery there at least on one occassion.
If you want to sample freshly baked Bosnian flat bread, called somun, I would recommend you go to the Alifakovac bakery. If I'm not mistaken, its the only thing they make here and they make it perfectly. The ovens get as hot as 500 celcius, baking the bread in about one minute, and the freshly baked bread smell is amazing.
While you are in the area you can also take a little walk around what is supposedly one of Sarajevo's oldest neighbourhoods and walk up the hill to the Alifakovac cemetery too.
Bakery Alifakovac details
Best place to get somun bread, a traditional Bosnian flat bread
This bakery is almost a bit of a secret. It's tucked in a very small courtyard just down a little alley in Baščaršija (close to the old clocktower), which is already in many ways a series of alleys itself.
This bakery is cheap and open very late, so it is the perfect place to stop off on your way home if you've had a few drinks in the evening and you're looking to comfort your stomach before bed.
If you're in Bosnia and you no longer want Balkan baked goods, then go to the French bakery. There are several branches and its the original bakery offering alternatives to more traditional breads in Sarajevo.
They have everything you would expect from croissants to baguettes and from apple tart to pain au chocolat. It is a bit more expensive than other bakeries in Bosnia and Herzegovina. They also sell coffee.
To be honest though, you can more or less go into any bakery and most of what is on offer will be the same as in any other bakery. There is a certain range of staples that most bakeries seem to stock.