The best sort of camera for almost any type of photography is an SLR. Even a basic entry level model will have vastly superior performance and image quality to almost any compact camera. That's not to say that you can't use a compact for landscape photography, and indeed they do have the advantage of being much easier to carry, but for the best results a DLSR is highly recommended.
As for what lens to use, that depends on the circumstances. The most popular choice is of course a wide-angle lens, usually with a focal length equivalent to 28mm or less, because they are capable of capturing panoramic scenes. However a good medium telephoto or long zoom lens is also very useful. The flattened perspective of a telephoto lens can work well for landscape shots, and is especially good for getting close to distant features. The best solution is a small selection of wide-angle prime lenses and a telephoto zoom. Personally I have previously used 28mm, 35mm and 50mm prime lenses along with a 70-300mm zoom, although I have recently bought a good quality 16-80mm zoom lens, saving weight and space.
If you're using a compact camera, you may be able to get a wide-angle adaptor for it, which will normally shorten the focal length by about 25 percent. If your camera can use these lenses they are a good investment, since the dramatically increase the versatility of your camera.
Since you will have to carry all your equipment to your location, a good weatherproof kit bag is also an essential item. Personally I prefer a rucksack-type bag, since it's a lot easier to walk long distances with your load on your back rather than slung over a shoulder. There are many different makes available, so shop around. Try to get one that includes some extra space for the outdoor essentials that I mentioned on the previous page.
You'll also need a solid but lightweight tripod. Landscape photography usually involves using very narrow apertures in order to maximise depth of field, which also means slow shutter speeds. Without a tripod you risk ruining your shots with camera shake, which is the last thing you want if you've had to walk five miles to get them. Again there are a number of types and brands available, ranging from budget models costing under £40 to high-tech carbon-fibre ones costing around £400, so get the sturdiest one that your budget will permit.
Before setting out on a trip, make sure to charge your camera battery and clear down your memory cards. It's a good idea to take spare batteries, especially if you're going out in cold weather. Keep one warm in your pocket while the other is being used, and swap them over from time to time to extend your shooting time. As for spare memory cards, high capacity cards are so cheap nowadays that there's no excuse for running out of storage space