Here's a good list of how to meet, befriend, and seriously get to know a girl. (Read it all. You'll get something out of it. Believe me, this stuff works!)
Steps
BE CLEAN. No one likes a smelly guy! Trust me. Brush your teeth, take showers (notice the "s"), and please wear deodorant. This is the number one step if you are to even approach an attractive girl.
HAVE A GOOD ATTITUDE. No girl likes to watch a guy put on a show, so be yourself. Don't try to be Mr. Popular and impress everyone. Girls dig humble guys.
SHOW RESPECT. If you respect everyone and give them their space, they'll all respect you. With girls, it works even better. If they see you respecting everyone, and not getting pushed around while you're at it, they'll think twice about letting you go on your way without saying hi.
HAVE GOOD CONVERSATIONS. Don't talk about how you beat the big-bad-boss on level 23 gazillion in the newest Barney Video Game. Most girls aren't too interested in video games. The #1 mistake when talking to a girl is focusing on yourself. Girls are way more comfortable when they talk about themselves. Ask about their interests, hobbies, favorite books, music, etc. If she asks you a question, answer it in a few, short sentences and then redirect the question back at her. It also helps if you don't have anything else to say.
All of the numbers above should get a girl to at least notice you. Now, it's time to notice her back (if that makes sense). If she is taking quick glances at you and you both make eye contact, do not, I repeat, do not be the one to look away! If you do, she will think you are timid and cannot hold your gaze. When you both make eye contact, just give a slight smile and, if you are cocky enough, give her a wink. She'll look away and will probably blush, but do not divert her gaze.
When around her friends, respect them and their ideas and opinions. Always speak highly of her and never, never, never talk trash about her. It will damage her reputation -- and yours.
BE FRIENDS WITH HER PARENTS. Be kind to everyone, especially them.
Learn to accept defeat. It's not always a bad thing. Take it and learn from it, so that the next time you see an attractive girl, you will know your weaknesses and try to build on those.
Tips
A REALLY JUICY TIP IS TO SHOW SYMPATHY. If the girl you really enjoy being around is sad or hurt, give her a hug. Don't just say that you're sorry and you hope she feels better, and don't just give her a 1 second hug (unless she wants one).
Hold her -- left arm around her waist, right hand at the base of her neck or holding her head. Let her cry on your shoulder and comfort her. Sometimes, the moment doesn't call for words. Just the sheer fact that you are holding her will make her feel a lot better.
Warnings
If you have followed all the steps above and it just doesn't work out between you and her -- and sometimes it doesn't -- remember that you tried and you at least (hopefully) got her attention.
If she dumps you for some other guy, it doesn't mean that you can never talk to each other again or that you can't be friends. Sometimes, good friends are better than a girlfriend or boyfriend.
If you hear that her and some other guy are together, be happy for her.
Don't be snotty and never talk to her again.
Ask how things are going between her and him, and if she really likes him.
If you show interest in her, even after you have broken up, she will still enjoy being around you because she probably knows you better and isn't afraid or scared that you will turn her down. If she enjoys your company, hang around.
Also, get to know her boyfriend. If you and he are friends, you will get to hang around the girl you love.