Studio Ghibli.
I watched Pokemon as a kid, like everyone else did, but that was just because it was Pokemon. It didn't spark any desire in me to explore the medium. At the same time there were several other similar shows on TV; Beyblade, Digimon, Yu-Gi-Oh etc. I watched some of those, but much like Pokemon, it was just because they were what was on TV at the time. It was the craze, it wasn't really anything to do with anime. I didn't even know what anime was at the time, technically. Around the time I also saw a bit of Dragonball Z, but I never got that in to it. It was shown at weird times for me, so it was a pain to watch, and the snail's pace at which the narrative progressed bored me as a child. I also didn't really have any idea who the characters were since I only saw one random series that was right in the middle of everything.
But then I started hearing more about anime proper, and I thought it sounded interesting. Then Film4 ran a series of Studio Ghibli films, and I was hooked. Loved them all (except Grave of the Fireflies). I went and bought loads of their films, and I've watched them over many times. Some I've even seen well more than five times over.
Around the same time (I think it was just before I discovered Studio Ghibli) I saw an anime TV series called Oban Star Racers. At the time I was just starting to understand what anime was, but I was still fairly ignorant. Either way, it was a good series that helped draw me in. I watched it again recently and was pleased to find it still holds up. I remember thinking it was the logical extreme of anime character design in the aspect that no-one had any noses. Or at least, not humans. It had a bit of the hair thing going on too, but I think Yu-Gi-Oh is a better example of going balls to the wall in that department. I've never liked that show though. To me it always felt like it was an anime being made by someone who didn't really know what anime was, or was trying to make fun of it.
But Studio Ghibli was what really started my love for the medium. I adore those films and regard Miyazaki as somewhat of a demigod. They're still fantastic after seeing them so many times over and after so many years. Princess Mononoke is still my favourite film of all time.
As time has gone on I've found a fondness for sci-fi/cyberpunk anime. I also tend to prefer series that are shorter and more mature than your usual series. For example I've never really been able to get in to Bleach, even after trying on several occasions, but I really like stuff like Ergo Proxy. In fact, I have the whole DVD collection coming to me in the mail right now. I aim to check out Steins;Gate sooner or later too. I saw the first episode and thought it was pretty good, if a bit baffling. Death Note is also another example of a series I like. It's shorter that usual (as in it's not 5 million episodes long and still on-going) and it has more contemplative themes than something like a protagonist becoming the ultimate ninja of destiny, or whatever.
Initial D is also another of my favourites. It's not very cerebral like the rest of the stuff I usually like, but it's a reasonable length. Not stupidly long, though. There's four series currently out with some extra non-essential stuff out too. There's two hour long side story specials, and some cut down re-done highlights of older episodes. I just ignored the latter. The story and characters are reasonably written, but the main reason I watch it is just because I love Japanese car culture. I've seen a few episodes of Wangan Midnight too, which I plan to properly get in to at some point.