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Also to add, that if their fathers at Mass are weird and not willing to talk, then you are going to have problems down the line, even if you can get married to the girl.

Any father worth his salt will show interest in young men who are potential suitors for their daughters. 

The others are usually insecure and manifest that through possessiveness. Go down that road if you want, but don't say you weren't warned.

Compliment her on her head-covering and modest dress, in a way that is chaste, gentle, and dignified.

For a woman to even take interest in speaking with a man, and to open up a little, a certain trust must be established, and even the appearance of a man will dictate to her whether he is worth speaking to. I am not referring to whether one is ugly or handsome, but to polite, pure manifestations such as appropriate grooming, demeanor, posture, etc.

Show by your dignified, modest body language that you are a serious, well-meaning man. Chaste compliments are a wonderful way to establish a little trust.

Contrary to what many imagine, women are quite taken in with a man who is genuinely interested in her welfare; that is, if she is not proud or impure.

As an introvert, it may be difficult for you to appear confident, so let your smile and posture be your confidence, and be genuinely charitable about it, desiring to bring a little light into the conversation.

Ask what her thoughts were on the sermon, and this is at the same time, a test of her depth. Share your own thoughts, and observe how interested she is in spiritual things.

It is always baffling to see how many traditional Catholics speak about useless subjects, right after having been the sacred tabernacle of Our Lord in Holy Communion.

One can ask questions about their patron saint, family traditions on Sundays, hobbies, to see if you share similar interests outside of the Faith. Favourite literature, secular and spiritual, favourite kind of music? Ask if she likes to cook, what her favourite dishes are to prepare? Share what you like to prepare?

Questions such as these will help you establish how much she cares about her religion, and how she spends her free time.