You need them both at the same time

Install the mat when you install the speakers. I would not expect much Difference in the audio system with just Mat, but it would be a quieter ride.
You need to install felt if you tint the front windows as well since there isn't any from the factory.
Another thing to think about is how you will power the new speakers since the stock HU is 2 ohm and most comps are 4 ohm, you can use a LOC (Line Output Converter) and amp or just get a new HU. Or new HU and amp :3eyes:
As you can see a simple improvement can add up fast.
Get a set of comps, some deadening and foam (if you really want to quiet the ride ensulite foam works better than deadener but does not dampen vibrations since it does not add much mass)
I'd suggest removing the factory plastic Baffle the 6x9 is mounted in and use it to design up an MDF adapter and mount that to the door with some dense weather striping tape to get a good seal and prevent vibration (if your sealing up the doors). Just need a Jigsaw and a pencil and place to work it doe snot take much time and gets rid of the factory "rattle trap" adapter for good.
for around 200 check out diamonds (as much as I hate to say it) or DLS MS6 (heard real good things about these they have magnesium cones). CDT's are ok from my experience but I have only heard their high end stuff.
Mount the tweeters in the dash if you want to keep it factory looking.