I just swapped out the speakers in my 2012. The 2013 is a bit different on the door speakers though. The speakers are a component and tweeter in an enclosure, and the mount doesn't fit aftermarket speakers.
What I would first suggest doing is finding a 2010-2012 Mustang door speaker mounts, either from a junk yard, online, or through Ford. That would make it a lot easier to mount aftermarket speakers.
Now, in my car I installed a set of Polk Audio Db5251s in the door using a Metra 6X8 adapter plate, and MM571s in the rear. The Db5251s are a pair of 5.25" component speakers coupled with separate 1.75" tweeters. The MM571 are 5X7 speakers, same size as factory.
The door speakers required a bit of modification to the factory mount. Specifically I had to use a utility knife to cut off the 4 verticle tabs, then cut a small U shape to fit the tweeter in the hole, and 2 triangle notches on the other end to fit the screws from the 5.25" speaker in the adapter plate. The plastic used in the mount is fairly soft and cuts easily. It only took about 5 minutes to get the speakers to fit without rattling.
The rear speakers are much easier, although you do have to strip much of the interior to get to them. Once there though, the 5X7 speakers will bolt right in.
BTW, both sets of speakers come with their own passive crossovers. I found the easiest way to mount them was to use velcro tape. I stuck the crossover for the door speakers in the top part of the interior door panel about even with the seat back, and the rears I just stuck to the bottom of the rear deck. Using velcro both provides cushion to prevent them from rattling, and it allows them to be easily removed should I ever need to.
I'll tell you right now that the sound quality difference is like night and day. Ford uses some cheap, nasty, Chinese made paper speakers that sound like the came from the 70s. Just swapping out the door and rear speakers with some really good ones will completely change the sound of the system.