Mostly I've just been lucky that I have awesome neighbors. It almost feels like college again, knowing so many people who live within walking distance. It helps that all of the houses were built within the past 2-3 years, so everyone is "new." I took a little welcome basket to the people who moved in on each side of us, and we've become friends with one side. (The others are nice, and they wave, but that's as far as they go.) And the people across the street brought us a gift when we moved in, and they've become friends, though not quite as close. Another friend down the street is just super social. And there are a couple of others that I just happen to have something in common with, so we text occasionally.
The neighborhood already had a Facebook page before we moved in, and that has helped us all feel like a community. I despise FB and only joined so I could be part of the neighborhood group, but it was worth it for that.
So I guess I would say, if you want to get to know your neighbors... go in person with a small gift to welcome anyone new (they'll be pleasantly surprised), start a FB page or other easy way for everyone to communicate, and maybe extend a no-pressure invitation (like coffee or beer or something, and for multiple neighbors at a time, like everyone on your block or floor or something, or the whole neighborhood if it's not very big). Oh, and/or you could use my mom's method... she's lived in the same neighborhood for decades and goes for at least one walk daily without fail, so she knows everyone and their dogs!
Beyond the neighborhood... thanks for the suggestions. I'm unfortunately not very sporty. Or not organized sports, anyway. I used to hike all the time, but there's no hiking where we live now, which is sad. Meetup is unfortunately not really a thing here. I have been to some Meetup events in the past, in another location, but they never met often enough for me to really get to know anyone.