I don't watch everything, but I watch a lot of sports. How I got into them though depends on the sport itself:
In no particular order:
Cricket is a family thing, played a lot
Rugby (rugby union, rugby league bores me to tears) due to it being mandated by my school but it grew on me, and it's also my dad's sport of choice
Soccer is an inescapable reality of growing up almost anywhere in the world other than the US, played a lot
Baseball I got into via cricket (very similar both mechanically and socially), played a little and some slow pitch softball
Football started as a childhood fascination during a family vacation in the US, ended up playing a little
Basketball was via friends who were already into it
Motorsports (mostly F1 but some others also) through my brother who was always very interested
I'm probably missing a few others, but nothing major comes to mind. I'm also a sucker for just watching a random sporting event every now and then; I'll watch the hurling grand finals, I'll watch the Aussie rules football finals, that kind of thing.
And yeah, I don't watch hockey. Sorry! It always felt, I don't know, gimmicky to me? It's 6 a side soccer... But on ice. And you use sticks. And skates. And not a ball. Oh and also just smashing the hell out of an opponent in front of a referee is just a thing that happens. I'm not explaining myself too well, but it's like someone took a fine enough sport as it is and just kept adding as many ideas as possible to it. Hockey, to me, is the sporting equivalent of too many cooks spoiling the broth.
As for teams I'm a fan of? Mostly it's the local team, but in cases where that isn't possible (the Mets, for example), it's due to family. For watching new sports I'm not familiar with (again your Aussie rules football's of the world), honestly it's just osmosis. Watch any sport for long enough and you'll pick up most of what is happening, even if you don't have the full nuance of everything. That will come later though!