I've received a lot of email from artists just starting out, and many have asked for tips/advice from me. I'm going to attempt to offer as much general advice as I can on this page.

Look at other artists' work. Look at all kinds of styles; not just fantasy art. By viewing other art, you can get a feel for how the artist works, and that will in turn help you improve. Like how one artist uses color? Like the lighting another artist does? Try analyzing it and try it yourself. A lot of artists, myself included, began drawing by copying other artists. That's fine for when you're starting out. The key is to gradually move beyond copying and developing your own style. The less you rely on just one artist's style in favor of drawing together bits and pieces from many artists as you learn, the better, as it will get you going on your own style. Don't ever feel like you have to conform to any given style, either. Your own imagination is honestly your best guide, and a personal style will just naturally emerge over time.

Try out all kinds of materials; pencil, brush, paint, markers, pastel, ink, digital, etc. My favored medium is a combination of markers and color pencils, but I've tried out most every other material available. Experiment, have fun with it, and again, don't get discouraged.

You don't have to buy expensive pencils or paper to draw. I start every drawing with a normal mechanical pencil. I used regular wood pencils for years, but switched to the mechanical ones for the convenience of not having to sharpen it. I'm not talking expensive mechanical pencils here, either. I tend to draw on smooth sketchbook paper, or, most of the time, plain loose paper. As with materials, though, if you can afford it, try fancier papers. There's all kinds of drawing surfaces available.

Drawing from life is invaluable. The more you can draw from actual people, animals, objects, the better. Drawing from life gives you a better feel for anatomy and form than anything else, especially when you're starting out. Photos are the next best thing. They aren't as good as live models, but they're a lot more convenient. Ask friends to pose for you, if possible. You can also use yourself as a model; especially helpful for hands and feet.

Art classes can help a lot, but they aren't essential. Personally, the classes I took working towards my Fine Arts degree helped me improve my art greatly, especially figure drawing classes; however, I know plenty of fantastic, professional artists that never took an art class.

How-to art books can help. I only have a few definite recommendations, but I'd suggest going to a larger bookstore like Borders or Barnes & Noble and browsing the art section.

For figure drawing, I highly recommend 3 books by Giovanni Civardi: "Drawing Human Anatomy," "Drawing The Male Nude," and "Drawing The Female Nude."
They're great reference/instruction, and they're pretty cheap at $15 each.

I also recommend "How to Draw Comics the Marvel Way" by Stan Lee and John Buscema. A great book for basic life drawing (albiet super-life), composition, storytelling, and inking. It doesn't matter if you aren't into comics or doing comics, it's a good resource.

For animals:

"How to Draw Animals" by Jack Hamm. Good basic animal drawing intructions.
"The Art of Animal Drawing" by Ken Hultgren. The author was an animator at Disney. He breathes life into animal drawing. Very high recommendation.
Photo books are extremely helpful for animal reference, too.

Again, browse the bookstores for more books and inspiration.

Another very helpful resource for reference pictures is Google Image Search. This is especially great for images of specific animals, armor, historical place, etc.

The above advice is based on my personal experiences. I don't consider myself an absolute authority on art, fantasy or otherwise. Hopefully, this gave you some things to consider and will help you out in some way towards your own artistic path. Good luck! :)