I am a bit confused. You say she says she is a shy person, but she has been the one initiating everything with you? If she is really shy and she is still initiating things with you, then she probably really likes you and wants things to go further. If she isn't really shy, she may just be saying that because she thinks you need to play the "man role" to make the moves on her. Either way, it appears she is into you like you are into her.
Instead of saying, "Let's go on a date," just ask her if she would like to see a movie and have dinner with you. Dinner and a movie is pretty standard date activity, so she should pick up on that. If she says yes, assume she knows it is a date and proceed accordingly. If she suggests bringing some other friends along, pull back a bit because she is definitely not ready for another step quite yet.
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