The preferred way to get to know me is to do something with me and let the conversation flow from that. Like most introverts, I detest small talk. Not because I’m some super brilliant intellectual who looks down on people who enjoy it, but because frankly I find it to be mundane, forced and awkward. Like most people, I don't enjoy feeling awkward. Plus, like many shy and introverted folk, it’s not exactly my forte. Extroverts who want to talk with me often find I’m an easier shy person to get to know than most. This wasn’t always the case. When I was younger I had no clue about social skills –I didn’t know how to start or sustain a conversation –I was also too anxiety ridden to talk to anyone above a whisper. Time, therapy, and social skills training have helped a lot.
If you are dealing with an extremely shy type of quiet person, you may be better off just being polite and non-threatening. That means NOT jumping all over that person with questions and chatter or being all weirded out because they can’t finish a conversation. They simply may not have the skills to interact in the way you want. Just let it be. Just smile and say something like, "I enjoyed talking with you." Do not be sarcastic. Be kind and sincere. No pressure. Over time, when they see that you’re nice and not some pushy extrovert who is out to make them feel horrible about themselves, they may open up to you.