The next essential is the tripod. If you want to take great landscape shots, you are going to HAVE to have a tripod. Keeping the camera still is paramount in landscape photography. You may notice that when you shoot a landscape hand-held, the horizon very seldom comes out straight. This is because the simple action of depressing the shutter is enough to dramatically tilt the camera.
A a general rule of thumb I always use a tripod for any shot with an exposure time of greater than 1/250th of a second and very often this is one of the demands of landscape photography. Many photographers won't take a landscape shot without a tripod, in fact.
Cost is less of an issue here.
Tripods can range from cheap 10 dollar models available in your local Wall-Mart to 800 dollar carbon fibre Gitzo tripods. There is a whole school of thought about what sort of tripod you should buy and trust me, there are books written on the topic. I'll have a huge tripod guide written for you at some point in the future.
For now, as a beginner, my recommendation is that as long as the tripod will remain rigid and steady whilst the head allows decent movement and locking then a $30 tripod can be as good as a $300 tripod, provided you actually use it.
In windy weather you can use a weight or even a couple of old bricks or large stones lying around. For rain there are plenty of covers available for both you and the equipment but I have used a cheap $25 pop-up tent for this with no problems.