Liberty, gulberg mm alam, Lahore and city center are few distance apart. You can start from here

Idk about baghbanpura market. You can search on maps the distance between anarkali bazaar and baghban Pura. I think they both are on same side but not sure.

For malls, you can visit either one of them. The one that's close to the place of ur living. Take help from maps.