Sarajevo is the city with mountains in the bus timetable. Trebević is the house hill. Bjelašnica and Igman are the Olympic day trips. You don't need to overnight on Jahorina unless skiing is the whole point. You do need decent shoes and a weather eye. Valley fog and ridge wind are different days.
City hills and Olympic day trips
Trebević cable car plus a walk is the easy win. Bjelašnica is a fuller mountain day. Winter, the same slopes are ski hills, though Jahorina has more resort infrastructure. Via ferrata is a guided add-on, not a solo experiment on rusty iron.
Using town as a base
Stay in the city for mixed days. Open the Book here tab for trips that meet in Sarajevo and the Olympic mountains.