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For about two years now, two of my bathroom fans have not worked and I haven't gotten around to changing them. But I am wondering the best way to replace it. The current fan/light I have installed is round like this 
but my new fan looks like 
I don't have easy access to the attic. Just looking for some advice on how to attach the new fan and the best way to create the new sized opening for the new fan.
If you have to do this from below then you need to remove a large section of drywall such that you can secure the metal housing for the fan to supports. Once you need to cut a large section of drywall you might as well mark the joists and cut back to 1/2 the width of the closest joist and remove from one joist to the other, for the other dimension whatever makes sense and add some nailing edges.
Ultimately you'll need to tape, mud, sand 4 edges and buy a full sheet of drywall. A little extra length to the taping isn't going to change much. You'll also need to paint for this kind of thing you might as well paint the whole ceiling.
Since you've indicated there is an attic above depending on your climate you'll have to patch or install vapor barrier and do a good air sealing especially in a bathroom.
