Anime...there are many great anime series out there.
I will always recommend "One Piece" for any one who wants to watch anime. It is my favorite anime. It started airing around 2000 (I think, based on my memory!) and is still on-going.
It is one of the best animes in handling "fillers" (i.e. story that is not in original Manga/script); unlike Naruto's fillers which are usually lame and kills the interest in the series.
Naruto and Naruto Shippuden are still recommended for watching (if you know the fillers' episodes, you can think of skipping them and to concentrate on the original story)
If you like swords/samurai anime; then I would recommend "Rurouni Kenshin" and "Samurai Shamploo".
If you are looking for sports-based anime, then: Hajime no Ippo (Boxing), One Outs (Baseball), Cross Game (Baseball), Eyeshield 21 (American Football), Slam Dunk (Basketball), Capeta (Kart racing/racing) & Prince of Tennis (Obviously Tennis!).
If you are looking for Robots/Mecha anime; then you might try watching a Gundam series. For a Super robot anime, you can try GaoGaiGar.
Other notable series that come to my mind are: Fullmetal Alchemist, Full Metal Panic, Bleach, Fairy Tail.
Hope this help; enjoy!