Those are just shells, if you weren't aware, which isn't going to do much for you unless the old kit's shells are badly damaged and out of round, or you particularly want a different wood sound. In the interest of saving money and not buying stuff you don't really need, if the physical state of the old shells are ok, and the hardware is fine, first just order your cymbals and some new drum skins.
You'll want to get new batter head skins and new resonant head skins, for everything, including the kick drum. (you would have had to do this anyway, so you might as well just do it first and see if that doesn't fix the issues.)
There are a lot of different drum heads out there, but in general, you'll want either Evans or Remo for any drum piece, and the same heads for all the toms. Toms typically get the clear skins, and the snare usually gets a white coated skin. For the drum bottoms, just make sure to get resonant heads for whatever piece, according to that piece.
Don't forget the kick drum skins.
Sometimes you'll find whole kit packs of skins. If you go that route, make double sure everything you need is in it, and the sizes are correct.
Your drummer will have to measure the shells to let you know what size heads you'll need to get. If he doesn't know how to do that, tell him to google it.
When the skins arrive, put them on the drums and tune them PROPERLY. There are millions of youtube videos that will walk you, or whoever is doing it, through the process. Doing this will give you the best sound possible.
New skins that are tuned correctly, plus new cymbals, will be a big step up. Any bigger of a step up than that will involve you buying a very expensive, high quality drum set, and not just the shells.
Let me know if you need any more help.
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