I just bought a D810, and I would greatly appreciate lens recommendations. I will primarily use it for landscape, wildlife, and occasionally portraits. However my priority at the time would be landscape and wildlife. I'd really prefer not to spend over $1,000, but I will if need be.
Quick answer as this is likely to be closed as a "shopping question" (which it is, and I'm not sure rewording can save it).
Your D810 is a full-frame camera, so only go for FX lenses. While you can shoot with DX (crop) lenses, you will be losing all the advantages of using a full-frame camera while doing so, because the image will be cropped to match the lens's projected image circle.
Secondly, because of this issue, a DX camera/lens is typically recommended for someone who likes wildlife shooting, because the cropping simulates a longer reach, and most wildlife tends to stay far away unless you are a phenom at field craft. To get a 450mm equivalency on DX, you only need a 300mm lens. On FX, you need a 450mm lens. And the difference in price is more than your entire budget, if you buy new and OEM. Because critters move fast, you will probably want a lens with AF-S for speediest autofocus.
Landscape lenses are particularly difficult to recommend because different people have different tastes and techniques in what comprises a landscape photo for them. They can be shot with ultrawides, wide angles, normal, and telephoto lenses, depending on the taste and abilities of the photographer. Most recommendations would probably say an ultrawide zoom, but they exhibit distortion and are not particularly easy to master immediately. If you have a 24mm-something or 28mm-something kit lens that came with the camera, that's probably your best starting point. If you can learn to use a tripod, to stop down for sharpness and depth of field, and to use a cable release, that will probably work better than immediately "upgrading" a lens you haven't even used yet. Bad technique will follow you no matter what gear you have. And it's you that's taking the shot, not the camera.
See also: Why are my photos not crisp?