As an otaku and a fan of japanese animation myself, I'm going to throw some suggestions. You should watch the Case Closed movies; they are incredibly good compared to the actual series, that becomes a bit repetitive after a while, not to mention the awesome thrillers in every single one of them.

There's also...
1) Azumanga Daioh - VERY funny Slice of life genre, it makes you laugh non-stop from events happening in our daily life. 2) Ouran High School Host Club - It's a Romantic-Comedy type, along with a bit of Slice of Life. It's still very good nonetheless. It's a bit for girls, but don't be ashamed to watch it, it's a laugh-tastic series for both guys and girls.

3) Neon Genesis Evangelion 1.0 - You're not alone(the movie actually). You're going to say; ''Mecha robots fights again?'' after watching the opening, but it's not only that, it's a great Sci-Fi series. The storyline somehow doesn't make any sense, but it's still good nonetheless. I'd recommand you if you like watching Drama filled with Psychological wounds along with great robots fighting scenes. It's a bit morbid, but still good. 4) Clannad - Very tragic story, it's enough to make several men cry with tears. Only watch it if you like real life drama and sad stories.

I can name some more, but I'll leave it at four, since I don't know if you have the same tastes as me. I'm more of a Slice-of-life oriented, and not a mecha robots-oriented(Evangelion being the sole exception for me, which means it gives you more than stupid robot fights; that's one more reason to watch it).

One more thing; the anime Inuyasha will leave you in a cliffhanger at the final episode, so I'd suggest you to read the manga instead of watching it. I finished all 167 episodes(or something like that) of Inuyasha, and I can tell you, the final episode was a big disappointement.