The difference between an introvert and an extrovert isn't necessarily that introverts don't like to meet people. Rather, while extroverts thrive on being in contact with a lot of people and maintaining lots of little relationships that aren't very deep, introverts prefer to engage people on a more personal level. In practice, this means that if an introvert and an extrovert go to a party, the extrovert will run around conversing with just about everybody for a few minutes, engaging with them but not really talking about anything of importance. The introvert will prefer to stay back away from the noisy crowd and talk with just a few people, but for longer periods of time and on a broader range of topics that go beyond the surface level. They prefer to get to know the people they talk about, whereas the extrovert is engaging with people but almost for the sake of engaging.
That's quite a bit of generalization, I know, but it might help give you some insight on what sorts of situations you should put yourself in so that you can engage people in the way that you are most comfortable with. So parties and large gatherings of strangers are probably not the best places for you to go. Try to find a club for something you're interested in--a book club, chess club, art club, anime club, whatever. It will be a smaller and more controlled group than something like a party, you'll be in a situation where it will be easier to have longer conversations, and you'll go in knowing you all have at least one thing in common. Once you make a good set of friends within that group, you'll start to meet their friends and it will evolve organically from there. Just remember to always strive for quality over quantity. And if they try to get you to do things that make you uncomfortable and won't respect your wishes even when you say you don't want to, then they are not friends.
And about your relationship question: I never got my first boyfriend until I was in college, but something that I think is pretty universal to relationships no matter how old you are is that you need to be prepared to spend time on it. If it's anything serious, you can't just expect to be able to relegate it to whenever you have a bit of free time. It's something you'll have to set aside other things for. Of course it should be fun and not a chore, but you'll find yourself having to balance "us" time alongside your "me" time whereas before all of your free time was "me" time, so just be prepared for that change of pace.