Adding fast rings will help a bit but actually sealing the door will make a bigger difference. Since you have to remove the door panels, might as well do both at once right?
One of the things you can do to the speakers when you do the install that helps tons is the speaker gaskets. They help the mid bass punch through the speaker opening and avoid it getting trapped in the cavity between the door skin and inner metal door. Definitely helps make things sounds stronger.
Road Kill RKFR6 3 Piece Foam Speaker Enhancer System Kit for 6" & 6.5" Drivers https://www.amazon.com/dp/B076MM4DJV/ref=cm_sw_r_cp_api_i_r4-qFbRCE78D5
A cheap way to get results is the look into improving the sound from your speakers.
You can add F.a.s.t rings on your speakers and/or add sound deadening mat in your door.
These should really improve the mid bass from your speakers for 30-100$.
F.a.s.t rings : https://www.amazon.com/dp/B076MM4DJV/ref=cm_sw_r_cp_apa_i_WOwZDb4JT6K0D
One more thing, some foam speaker pads are also great for sound quality, something like this helps way more than you'd think it would. https://www.amazon.com/dp/B076MM4DJV/ref=cm_sw_r_apan_i_N5RWM880GH01WWDZB114?_encoding=UTF8&psc=1
So one major point of warning to start: If you have a "factory upgrade" audio system like Bose you generally cannot just swap out speakers. Usually those systems are built in a way that all of the components are made for each other and are not compatible with aftermarket components.
So I usually steer clear of those bucket style kits for that reason. What I usually use instead is a combination of Fast Rings, homemade brackets, and speaker gasket tape to mount the speaker to the door.
As a simple way of sealing the door I use CLD. There are better ways to do it but CLD is easy and effective. Your basic goal is to seal the door as well as add rigidity to the sheet metal to create a good pressurize-able sound chamber. You'll never get that door truly air tight but the closer you can get the better. For the most part that means a layer of CLD on the inside of the outer door skin (inside the door) and then a layer on the outer side of the inner door panel (facing the inside of the car). You can do multiple layers but there is a serious diminishing returns so I usually don't. You also want to keep the sheets as big as possible because it works better that way (more rigid).
Here is a post of pics from when I sealed my doors with CLD. I did a fairly sloppy job because I bought sheets of CLD instead of a roll. Buying a roll means you can do the entire door in just 1 sheet which looks better and also works better. You'll notice that there are still the small holes where the panel pops into place, can't do much about those. Also I didn't do anything to the bottoms of the doors on the exterior but I did line it with CLD on the interior. I have not had any issues with water retention on the interior of the door using this method.
https://www.reddit.com/r/CarAV/comments/f4zznu/two_days_later_the_doors_are_sealed_ish/
Here's a cool trick that proves this all makes a difference. Do it to one door only, then close the door and knock on it lightly with your knuckle. Then knock on another door that hasn't been treated. The sound difference in amazing! The treated door will produce a low dull thud while the non-treated door will have a louder tinnier sound to it. Now imagine if it was a speaker on the inside producing the sound and not just your knocking!
Fast Rings:
Second Skin CLD
Gasket tape:
Cloth tape:
Replacing your door speakers yourself is easy as hell. Buy the speaker adapter rings, this will let you just pop them back in with very little fucking around. It's also worth buying fast rings. I'll leave you links at the end of this comment.
There are two t30 screws.
One in the door handle behind a piece of hard plastic (use a flathead to get it off) and one underneath the small rubber liner in the little box on the door by your window buttons.
Take the little piece of trim off the sail by your mirror, there are two clips here.
Then place your foot on the bottom of the door, grab the door trim by the little storage part along the bottom and pull hard. It will come off. There are just a couple clips that you can take off if you want. Or you can leave them on, just be careful you don't leave the door trim hanging or you'll break something.
It takes less than half an hour to do this.
Video on how to remove door trim
Wire colors (it's for 2012 but they are the same for all gen1)
As an aside, it's worth replacing your door trim retainer clips if you wanna spend a little extra. Once you pull them off, they are never quite the same and you'll likely break one or two of them.