For getting started in anime, I suggest looking over the genres below and watching the anime recommended
Fighter – Soul Eater. My recommendation for fighters is Soul Eater. Soul Eater is about a group of students at Death Weapon Meister Academy. Their goal is to upgrade their weapons to death scythes by consuming 99 human souls and one witch soul. Soul Eater is heavy on the comedy, with a lot of oddball characters and often is just plain silly. It also features a unique art style, melding gore and old time cartoon.
Giant Robot – Code Geass. Code Geass is about Lelouche vi Britannia, a Britannian prince who has been exiled to Japan. Lelouche gains the ability to make anyone follow a single command he gives, which combined with flawless tactical abilities turn him into a nightmare for the Empire of Britannia. Code Geass has lots of great action, from one on one robot duels to large scale battles. It is also not short on humor as Lelouche balances his life as a terrorist with his cover of being an everyday high schooler.
Harem – Love Hina. Harem anime typically feature a relatively useless guy who has a multitude of pretty talented girls fall in love with him. Its no different for Keitaro, the protagonist of Love Hina. While studying to get into Tokyo University, Keitaro ends up becoming the manager of an all girl dormitory. While they struggled with it at first, eventually all the residents develop feelings for him. Love Hina is primarily a comedy, as the girls endlessly torture Keitaro for clumsiness and stupidity.
Magical Girl – Sailor Moon. Sailor Moon is an iconic anime from the 90s. The titular character, Sailor Moon, is a staunch defender of love and justice, protecting the Earth from the evils of the Negaverse. Sailor Moon and her friends, the Sailor Scouts, transform from ordinary high school girls into their super powered forms via shiny makeup. Its a bit corny, but its still highly entertaining to watch.
Romance – RahXephon. RahXephon is a story of love lost through time… and an alien invasion. Technically, its a giant robot anime too, but the romance is a major focus of the series. Its a series filled with intrigue, mystery, and mythology. It was my favorite anime until someone else showed up…
Cyberpunk – Ghost in the Shell: Stand Alone Complex. Ghost in the Shell is about anti-cybercrime police unit known as Public Security Section 9. The series mostly focuses on the unit’s hunt for the Laughing Man, a cyber terrorist, but there are other criminals that put a stop to. The show is well thought out and the AI tank mascots are just adorable.
Space Opera – Cowboy Bebop. Cowboy Bebop is about a crew of bounty hunters working the beat in the solar system. Cowboy Bebop was an emotionally powerful experience for me and I have only been able to watch it once.
Overall – Full Metal Alchemist: Brotherhood. The creme de la creme. Fullmetal Alchemist: Brotherhood has seized the top spot in anime for me. The show is about two brothers looking to use the magic of alchemy to fix their broken bodies. There is good mix of action, comedy, mystery, philosophy, and politics that always keep you on your toes.