If you're really shy, don't try to "pick up" girls. Successful "pick ups" take a certain kind of personality and if you're really shy, it's not going to work out for you to just try going up to girls, having a casual chat, and getting their numbers.

Instead, just try talking to girls, making friends with girls you like, and building that relationship. Ease yourself into it. Some people jump straight into the pool and other people test the water first. There's nothing wrong with going either way, but if you're shy it's probably a lot easier to test the water first.

So get to know some girls, and since your goal isn't to get her number or score a date right away that should take some of the pressure off and help with your shyness.

When you get to know her better, it will be easier to get her number sometime. You can try inviting her to something and then asking for her number as a way to contact her, or just offer her your number if she needs it (there will probably be some occasion when you can pass this off).

I hope this helps.