I may be a contrarian here, but I think the D80 is a perfectly fine camera for serious landscape shooting. Are today's cameras better? Yes, in subtle ways, but if you're looking for best bang for a limited buck, specifically to improving your landscape photography, it might not be the best place to start.
Equipment-wise, landscape photography will want an excellent tripod, ballhead, remote release, and a complement of lenses from ultra-wide to telephoto. Past that, experience will be your greatest asset to improving your images, and that probably means some amount of travel budgeted in as well.
Just adding a good tripod, ballhead, and ultra-wide lens to your kit, along with a good backpack, will probably set you back minimum $1000, and that's on the conservative side. If you go the D600 route, the conservative approach is probably pushing $5000 ($2000 body, $700 tripod+head, $1200 ultra-wide, $700 mid-range zoom, $200 backpack).
I don't know where your budget actually stands. I'd certainly prefer the D600 kit to the D80 kit, but if that doesn't leave you enough money to get out to some great locations to shoot, then it's not really a path to serious photography, just to buying more equipment.