To answer the easy question first, I hand-draw most of my maps using the time-honored method of pencil and paper. I've been drawing random maps for years this way. For a big, rough sketch to have available on the computer, I'll use boring old PAINT.

As for the depth of world-building, I tend to stick to Tolkien's advice "make the map before you write the story". I don't generally have coasts perfectly defined, but major cities are placed, and rivers and other such things.

Races, a rough sketch of the history, and a few sketched religions are the first things I work on, after the geography. You don't need to go in-depth on your history OR religions, as each culture tells history differently, and religion is pretty nebulous as well.

Other than that, a grab-bag of related names, a few little places on the map, some legends and maybe a god or two, and it's enough to start writing. You'll find a lot of it gets filled in as you write.

I actually created my world based on my characters. I find that by knowing what kind of people and creatures live in that world you can develop the world that way as well. Granted some methods work better for other people. I just always start with characters, once I get to know them I feel like the places they're from and what they would be like is just intuitive.