What others are saying is accurate, but there's a little more that could be said. I've tried leaving a UV or haze filter on the lens and I've tried shooting without one, and I must say, there is such a small difference, none of your friends and relatives will know. The Tiffen HAZE-1 gets good ratings for cutting through haze, but I don't see much difference with mine.
As for lens protection, the only thing you would be protecting is the coating on the front element. If one is quite careless the optical coating can be scratched, as well as the glass itself. So a clear filter in front of it would protect against that. But that's all. As others say, if you drop a lens, the least of your worries would be a little scratch. I had a camera slip off my shoulder last winter (my big coat was slippery I guess) and it landed flat on the bottom of the camera. I looked at the front element, and it was fine. I looked through the viewfinder and it was fine. It wasn't until I looked into the lens that I saw an internal element shattered about halfway around the edge. I hate that feeling...
So a filter isn't magic protection- it's just a little insurance against front element scratches.
For a beginner, sure, try a UV/skylight/haze, you might like it. For really cutting through haze in landscape shots, try a circular polarizer. It also does nice things to washed-out blue skies. But be careful- you can turn the sky too blue, especially at higher altitudes and when underexposing.
Also, depending on your level of experience, you might like an 81-series warming filter to punch up drab scenes and get rid of the bluish cast landscapes can have. If you're a film user, you also might want to see if your preferred slide film has a color bias to it. Fuji Velvia and Provia (any others?) are notoriously blue, so a warming filter is often a must. Some say the Kodak E100VS is reddish or brownish, depending on who you talk to.
The basic thing to remember is that when you attach a filter, you are putting something in front of the lens that may or may not be of the highest optical quality. A crappy non-glass filter could degrade your image, while a very expensive pro model shouldn't affect the resolution. Also, be careful not to stack filters- or at least not too many! You will vingette the image.
The filter world is large, and only you will know what you prefer with experience. I use only UV/haze and a circular polarizer- maybe I'll get a warming filter soon, but I'm in no hurry.
Good luck!