I remember channel surfing (probably in the late '90s as a tween) and coming across Go Fish. Specifically, it was Xander's cafeteria scene where Gage turns into a fish monster. I didn't even know what I was watching. Years later, I recognized exactly what it was! I had grown up on Goosebumps, Are You Afraid of the Dark? and So Weird, not to mention a lot of old classic horror and '90s kiddie flicks with supernatural themes, so genre fiction did appeal to me.
But the first full episode I watched was Smashed when I was babysitting some tween girls in 2003 who had the DVD sets. I wasn't impressed at all (kinda grossed out). I was 16 at the time and I wondered why the hell the girls (10, 11 and 12) had these DVDs. In 2003, I actually was reading a lot of Anne Rice (first four books, though I only really liked the first) and liked the IwtV film. I started hearing about BtVS more because of liking the vampire genre, but it felt like everyone online was just talking about Willow's sexuality, which didn't exactly sell me the show any further than seeing Smashed had.
Then, the next summer (2004) when I was 17, I had my Internet taken away and so I ended up watching a lot of TV. I ended up marathoning the shows in syndication (I was taping them on VHS!) for the next half a year on FX and TNT until I bought the box sets. The Yoko Factor is the episode I first started watching (seemingly completely missed Angel--then never saw Fool for Love and missed a few other episodes, as well) and so I basically saw BtVS very backwards. Because I had no Internet access, I had no idea there was a spinoff and I randomly ran into AtS when I spotted Spike in Hellbound. I hadn't even met Angel yet in the BtVS reruns. Thus, I ended up seeing parts of AtS season 5 before the rest of it. AtS, however, did loop around well before I got to see BtVS seasons 1-3.
My 'shipping preferences completely swapped, as did my favorite character.