Books

For manga:

Ghost in the Shell and Appleseed by Masamune Shirow are very influential and well known (in the west) scifi manga, but they are pretty dense and technical and not everyone's cup of tea. Both go into a lot of very detailed world building and politics, but also have some of the coolest Japanese cyberpunk action every put to page.

Berserk by Kentaro Miura is probably the most famous fantasy manga in the west. Unrelentingly dark and grim, it isn't to everyone's taste but the spectacular art and unique, epic story make it stand out in the genre. Contains some nasty sexual violence.

Fullmetal Alchemist is a steampunk fantasy setting with great characters and story. It's aimed at a younger audience but mostly avoids the common tropes and cliches of that style. Easy to read and charismatic series.