Welcome to the forums
K-5 is a great camera, I have one and I love it.
There are many threads about the lenses here, what I've learned from them is that any lens can be used for landscape photography, some people shoot landscapes with 200mm lens others with a 10mm, I guess it is all about how you compose the shot.
First lens I would recommend is a wide-to-normal zoom. Most people here recommend Tamron or Sigma 17-50mm f/2.8 ($300-400 used or new). Both seem to have good IQ, Tamron is cheaper, Sigma has Hyper-Sonic Motor which is silent and would complement very quiet shutter on your K-5, some people also say it is faster to AF than screw-driven lenses. You can also get a Sigma 17-70mm f/2.8-4, DA 17-70mm f/4 or the 16-50mm f/2.8, it depends how much money you want to spend in this focal range. You can of course opt for several primes (15mm, 21mm, 28/30/31/35mm, 40/43mm, 50/55mm, 77mm...) which will give you better image quality and better low light performance, but that's even more money and more weight in your bag.
Second lens I would buy if I were you is something that would give me more reach. People praise the DA 55-300mm lens ($200-300 used) for it's image quality. If you need something faster you can get Tamron or Sigma 70-200mm f/2.8 new or used ($600-$1300). You can also add a couple of tele converters to that (1.4x or 1.7x and 2x) which will give you more reach when needed.
If you are OK with the idea of having one prime I wold look at DA 21mm f/3.2 ($400-500 used), FA 31mm f/1.8 ($750-900used), FA 35mm f/2 ($300-400used), F/FA 50mm 1.7/1.4 ($200-300used).
Some of the lenses may have front/back focus issues, others may have problems with SDM.
P.S. Manual focus lenses will cost less
P.S.S. I would also strongly recommend you getting a good tripod.