How do I remove this old bathroom exhaust fan?
I am assuming that new fan is bigger or same size.  


You have joists that this is nailed/screwed into.  Try not to damage adjacent drywall.  I am going to guess from this picture your joists are on the top/bottom of picture.
This is because your exhaust/and electric look to be coming from the left.  I would cut straight along the top and bottom (using picture) of the fan cut-out and extend that about 2 feet to the left. 
Remove drywall
Take out old fan with hammer or screwdriver or just club it out gently.
Install new fan - with the 2 feet you have room for maneuver duct into fan slot and electric work.  
Put new piece of drywall on and then you have 3 pretty easy seams to mud/tape.