When we moved we did a lot of things - extra cakes on bake day (we made too many cakes, wondered if you would like one?), same with jam, homebrew, etc.
Occasional interest in what they were doing outdoors (whats that mower you are using? A Ransomes tiple? Whats that?)
Oh....and in the summer - the barbecue. A classic.....
"Having a barbecue on Sunday for the neighbours....please come along".
This is now a tradition in our huge town of 5 houses and the farm. Everyone has one, but everyone brings stuff. Booze is all home made, as are the cakes, salad is home grown, meat is home reared, I grow the mustard, etc. Everyone has fun and its free (sort of). Takes time to get to that point, but you have to "put yourself out there". Its hard and can be a little nerve wracking - but really, wahts the worst that can happen? I hate the idea of "community" it smacks of a cardigan wearing social worker! But good neighbours and local friends are great to have.