IMHO best way to explore any place is on a bicycle if you can, in urban areas you need to get up early when the roads are free of traffic and just stroll on streets if you happen to be resident of this city then you can visit different places at different times. If its a group you guys can always have a breakfast GTG at the end of ride.
I am a bit puzzled about the kind of suggestions pouring in a thread for places to visit in Karachi, the thread has become somewhat related to places of interest in the entire Southern region of Pakistan, if it is so then IMHO crown jewel of south should be astola with its pristine beaches and amazing overall landscape, sorry for poor image quality they were taken with a mobile camera in 2009 I think and one shot is from planes window when we were coming back.

