Prices from B&H (you might be able to get it cheaper elsewhere). And from your description, you're not interested in tele (like a DA55-300).
If I would only spend it on lenses:
Rokinon 16/2 ($380), DA35/2.4 ($180) and Sigma 70/2.8 macro ($450); $10 over the budget
Lightweight for backpacking? Probably not. Versatile? Absolutely. All three are suitable for landscape photography, the 35mm will make a reasonable general purpose lens (I personally find it a tad too long, hence the '
reasonable') and can be used for full body protraits and the 70mm allows to pull things a bit closer, do head / shoulder portraits and adds macro option.
But my approch would be a tad different:
Up my game with a tripod and a flash first. That will be roughly $350 - $400. Add a FA35/2 ($400) and from the rest buy a decent ND filter or wireless flash triggers.
Always nice to spend other peoples money