A friend of mine got me to go to Otakon with him in 2002. I'm not exactly sure what the first anime I saw was, but I got reeled in because of FLCL and then the addiction started with Outlaw Star.
I have watched a lot of really bad anime, but I am entertained by most of it anyway so thats ok with me. I only rate things badly if it was not good AND I didn't enjoy it. I prefer the newer styles of animation, the prime example being Busou Renkin. I will watch things in other styles, but if its too goofy and cartoonish looking, or if its extremely realistically drawn, I usually don't like it as much.
My top 5 are not in any particular order, and are subject to change based on my mood. Busou Renkin and Fullmetal Alchemist (holy crap second series coming?!?) will probably stay in my top five since I can't think of anything about either series that irritated me, which is extremely unusual.
I also read manga when I am not feeling quite so lazy, or someone points out one that they think I would like. Some long running series are in my opinion much better in manga form, since there is much less fluff (for example Naruto... don't like the anime, but the manga is very good). Read Battle Angel Alita, it is epic.