Spending a chilling time in Sarajevo is not a big problem. VISIT SARAJEVO is you ultimate guide to chilling places all around this marvelous city.
First on the list of places to visit, if you like oriental kind of interiors and exteriors, is Dibek. Place in very interesting oriental design, with very interesting and unique drinks, teas, coffee, shisha, etc. Very much worth visiting, especially in the hot sunny day since the exterior there is very nicely decorated. Perhaps try magic tea (čarobni čaj).
In the same yard but just across another oriental design coffee bar Halvat, also very nice daily chilling place. Nice Black Elder juice they have. Give it a try.
Cafe bar Sova is a brand new open and offers very interesting interior, products, comfy, and great service.
Another tow café bars of oriental design Baghdad and El Kasbah. Two places in the old town of Sarajevo very much worth visiting.
In case you are not much of oriental style cafés than maybe try seaside motivated café bar Barka, old traditional wood detailed Barhana (not a pure bar, much of a restaurant, but great chilling place).
In the Strossmayerova street, the most popular coffee bar street in Sarajevo, you can give a shot any of the café bars there, such as: Central Café, Café Kiborg, Café Quadro, etc.
We mention that those are just basic ones, a bit common kind.
Again list of modern kind of chilling cafes in Sarajevo, but definitely worth visiting include: Vatra, Alfonso, Ort, Delikatesna Radnja, Aquarius, Incognito, Pravda, Von Habsburg, etc.
Each and every one is clickable and reviewed, so see what people say. Enjoy.