For the first family cat Max, my dad found him at his workplace (he works in a junkyard). He called my mom during lunch to ask if she would like a cat, and she agreed haha. Sadly he died a few years later as he wasn't feeling well and we couldn't afford to visit a vet. =<
Second cat, Ocean, my mom was visiting her sister and she kept seeing these kittens on the streets. She wanted one of them, so she and my cousin tried to catch her.
Our next set of cats went similar to Cool Porygon's story. Family and I moved into the house, and a stray cat would visit often. She ended up pregnant with three kittens. One of them we gave away while we ended up keeping the mother and other two kittens, Vanillia and 'Tom and Jerry'. Fast forward a few months later, and mama cat ran away for a few days and ended up pregnant again (we did plan to spay her, but then loads of emergencies happened and we used the money for those instead). The next kitten's named was October because that was the month he was born heh.
(Out of the mama's kittens, Ocean got along with 'Tom and Jerry' the most, lol. Ocean and mama cat got along fine for a while until mama cat was carrying October and then they needed to be separated in different rooms).