You should use the tips of your fingers, both hands. Left hand reads the first half of the line, then right hand reads the rest while your left hand goes back for the first line. This is the best way to increase braille speed. Think of the curve as how your fingers look when you're typing on "home row." Learning by sight is much easier but eventually you'll want to read interpoint braille which is written on both sides of the paper. You won't notice the dents (dots on the other side of the paper) by touch but it's tough by sight.
The best way to learn once you know the basics (including a few contractions) is by reading and using context clues.
$1 is a great site for books on sale. Don't try to read more than a ten or twenty page children's book, preferably one you've read before. It's really quite a bit more difficult than you'd anticipate. I recommend "The Magic Finger" by Shel Silverstein.