The most famous square in whole Bosnia and Herzegovina, Baščaršija square, and its sorroundings (shops, bazzars, souvenir makers, food and drink producers) are something you can not miss.
The focal point, Sebilj, most famous fountain in Sarajevo as well as whole Bosnia and Herzegovina.

Baščaršija and Sebilj
Once the biggest library in Balkans, Vijećnica Library, today the most representative heritage from recent war, which hopefully will not be for a long time, since it is about to be repaired, and about to gain back former bright shine, which it deserves. She serves the knowledge, and the knowledge is power.

Vijećnica, City Hall then Library
Next on the list Gazi Husref bey’s complex, and his list of endowments. Along in this complex you should NOT miss:

Gazi Husrev beys Mosque
War history, Tunnel of Hope – War Tunnel. Only connection of Sarajevo and free world during the Siege.
Probably the oldest and most-extraordinary (by its architecture, and nature) Orthodox church in Sarajevo, the Old Orthodox church of St. Archangel Mikhail and Gabriel.
Also do not miss the Orthodox Cathedral, one of the most impressive temples of Orthodoxy.
Next on the list, Cathedral of Sacred Jesus Heart.The most representative monument of Christianity in Sarajevo. Jewish people, history and tradition.
Jews have come to Sarajevo, I would say long time ago, in 16th century, and in the Ottoman Empire were equal to Muslims, so that is why they were free to build such an amazing temples and structures in Sarajevo.
Here we mention those most important ones:

Il Kal Muevu
Museums like museums, each and everyone has a story.
National Museum, the first of its kind that you should visit, after that, Historical Museum, Jewish Museum, or Museum of Sarajevo 1878-1918, Svrzo House.
Bridges, oh my God, we love them. They connect people, and people’s people.
First on the list, Latin Bridge, you know, Assassination, Franz Ferdinand, World War I. Unavoidable.

Goat's Bridge
Next, Goat's Bridge, Constantinople road, unavoidable as well; According the myth, built by the unknown architect, in the Miljacka canyon. Very interesting place, bridge almost forgotten.
Roman Bridge, nobody knows when precisely it was built, old around 460 to 480 years. Built of a stone, that was once used to build Roman structures. Highly recommended.
Also do not miss the fortresses, which stands near the city sky and conducts the four centuries old command to keep the enemy out.
Two in charge Yellow Bastion and White Bastion.
Sarajevo Roses, unfortunate heritage from the recent war. Click the link and see what they really are.