New Drumset and I know Nada about 'em
 

My drummer needs a new kit. The one he has is absolute garbage. The cymbals sound like tin cans and the drums sound like farts. :| Anyway, I'm going to help invest in a new one. I was thinking of getting this PDP Z5 5-Piece Shell Pack: Shop Drums & Percussion & Other Musical Instruments | Musician's Friend and this Paiste PST 3 Essential Cymbal Set 13/18: Shop Drums & Percussion & Other Musical Instruments | Musician's Friend and using the other sets hardware to set it up. Our combined budget is limited to around $500. Yeah its CERTAINLY not going to be a great kit but, hopefully better than what we have. Is the above a good idea or do you have any suggestions?

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.

One of the biggest problems with the set is the rims don't fit well. On one of the toms it doesn't even line up with the holes where the screws go. The Floor tom can't be tuned well because the rim is slightly bigger than the drum and it just slides over it. That was the reason we were thinking of a shell pack.

You'll definitely want a new shell pack then. I just wasn't certain if the particular pack came with heads or not, and if your original shells weren't bad off, new heads would have been the way to go if you were going to have to buy them separately anyway.

Well, that PDP may not be the best in the world, but I'm sure it's a leap over the current kit, so I'd say go for it, unless you can find some good condition second hand shells of a higher caliber brand for the same amount of money or less. If you did that, you'd probably want new skins, but it'd be worth it anyway.

Pdp makes some really nice kits in their upper end.

May not work, I'm not sure. I'm not a drummer. But couldn't you just get some new rims for the junky drums and new heads? That would be cheaper than getting new shells wouldn't it?

It's likely that the old shells themselves are out of round (warped), which would mean new rims and heads also wouldn't fit and tune properly as well.

Gretsch make really nice kits if you're willing to pay $700 or more on the shell kit alone (includes heads). You might just wanna invest in something more expensive because that'll take you farther in the long run.