This is gonna be a little different. You probably won't be a fan of this, but I thought I'd give it a shot anyway. :p Why not try out a Kamen Rider series ? Technicially it's Tokusatsu, but it's kinda anime based in a way. I'd recommend Kamen Rider Kabuto or Kamen Rider 555 for first time viewers (Really good in terms of action and story). Reason I recommend this is because alot of the anime now a days pales in compared to the anime in the 80s and 90s. Heck the last awesome anime I've seen in the 00s was Gurren Lagann. KR kinda gives a nice break from the same old anime that has been repeated over and over again. Just adding my 2 cents. ^_^
P.S. Even thou the series is mind fuck, Evangelion is awesome. :D