Seconding Akira and Ghost in the shell, Steins;Gate is great too.
One of the first ones I saw (and I love cyberpunk and scifi shit) was Serial Experiments Lain and it was one of the first ones to get me absolutely hooked. Mind you, it is weird and strange, but if you’re into stuff like David Lynch at all you’ll love it.
Others to look into that aren’t exactly cyberpunk, but futuristic/scifi nonetheless: Neon Genesis Evangelion (finish it, it gets really cool towards the end and I thought all things mecha-anything was just dudes being silly about robots), Cowboy Bebop, Ergo Proxy and Gantz (might seem silly at first, but gets interesting towards the end).