If you call up a shy girl out of the blue, it's possible that she'll give you an excuse to avoid going out. This isn't because she doesn't like you, but because introverted or shy-types like to have their plans in place. Shy girls need a chance to get ready mentally for a date.  If it's 4:30 on a Friday night and a shy girl doesn't have plans, she's probably looking forward to a quiet night at home. Make your plans ahead of time to give her enough opportunity to prepare. In general, try to make plans at least 1-2 days ahead of time at least. It's good to plan an activity for a date with a shy girl, so you'll have something to do together. If conversation is going to be hard, try to pick an activity that'll let her loosen up and have a good time, without having to do a lot of talking.  Going for a bike ride, bowling, or playing mini golf might seem like silly ways to go on a date, but it's a good way to keep yourself active and busy. You'll have a fun shared experience. Dinner is a good opportunity to talk, but it might end up being pretty awkward if you have a hard time talking together. Alternatively, movies mean you can't talk at all. Not the best choices. Make sure whatever you do is quiet enough for her to talk if she wants to. If you go to a booming club with loud electronic music, you'll never be able to talk. Shy girls aren't generally big partiers. If you ask a shy girl if she wants to go to a big dance, or to the football game, or to a loud dance club, don't be offended if she'd rather stay in and watch Netflix on the couch with you. Make the time that you're together special, even if you're not doing something that sounds like a date. Cook dinner together, instead of going out to a fancy new restaurant. This gives you the chance to do an activity that will get you talking, even if you're just talking about the chicken and rice dish you're making together. Shy girls are often intimidated by large groups, but you can always ask her about her preferences. While it might sound great to you to hang out with all your friends and watch football together in a huge group, it might be her worst nightmare. Try to give yourself some private time.  Of course, some girls might be more comfortable in group dates and feel more shy one-on-one. All girls are different. Talk to her and find out what she prefers and do what makes her feel comfortable. If you do go to a dance or some other kind of social event, come up with a signal that she can give you if she wants to leave or pick a designated time leave. Being shy doesn't mean that she doesn't have a strong opinion. When you're going out, give her the opportunity to pick the activity. This will help her feel comfortable and confident that you enjoy being together. Give her the opportunity to open up. Don't be afraid to take charge, though. Some shy girls might give you a lot of "I don't know" answers when you ask what she'd like to do on Friday night. Have a plan in place, if she doesn't. You should also offer more than 1 option to ensure that she can pick what she feels most comfortable doing.
Make plans ahead of time. Plan fun dates that will get her to open up. Spend your downtime together too. Make your dates one-on-one. Let her pick the plan.