What's a good anime to watch

I agree it depends on the Genres you like. Anime is a format, not a genre, it spans all genres drama to comedy to fantasy, Sci-fi, Horror, Thriller Mystery etc.

Me personally, some of the best anime have been original, not based on a Manga and are short.

Serial Experiments lain
Cowboy Bebop (This one is a no-brainer to suggest)
El-Hazard The magnificent World (OVA, not TV series)