Anime for a newbie?

I'd recommend avoiding long running big names at least at first: So No to Naruto, One Piece, Fairy Tail or Bleach. Even though these are Renowned and critically acclaimed Shounen animes; they do run very long. If you don't have time to marathon them; you may not be able to get into them...and a portion of the storyline may bore you.

If you really want to watch something longer running than one or two seasons then Inuyasha is a decent Shounen (Male Protagonists/Fantasy/Powered) but can be romantic like and viewed as a sort of a period drama as well. The original Sailor Moon series is good if you're into Shoujou (Girl Power/Fantasy/Magical/Romantic). Cardcaptor Sakura is also a good way to get your feet wet; and it's a deep enough story with very entertaining characters. But it is a shoujo; so if you're one of those guys who recoil at girly things...then you're gonna have to try another such as Death Note, Danganronpa (both are Murder Mysteries) or perhaps even Yu Yu Hakkusho (Mystery/Shounen). Tenshi Muyo animes are usually decent too, they're called Harem (Yes, harems of ladies!) Animes; as are Gundam (Mecha, Giant Fighting Robots, an Anime original thing~!) animes or even your random quirky anime like the ones you may find running on Cartoon Network's "Adult Swim". I think Samurai Champloo is one of them. (But those are totally not my thing)

If you like Pokemon; I strongly recommend the Digimon Trilogy; which is Digimon Adventure 1 & 2 as well as Digimon Tamers. Seasons of Digimon after that point are not as great as the series changed hands a few times after Bandai dissolved into Namco, and was licensed out to others. Digmon Adventure Tri is also out; which is a sequel to the Original Digimon Adventures 1 & 2; and it has been good as far as I can see.

I do strongly recommend Digimon if you're into Pokemon's Anime a lot. Digimon is more globally appealing as there are multiple, and diverse, protagonists. You will like at least one of these characters most likely

If you don't mind English Sub, I would highly recommend using ChrunchyRoll, which is an official anime watching site.

I would highly suggest ERASED, Food Wars, Sword Art Online and Parasyte and Akame Ga Kill. Although the last two can be pretty graphic.

I don't like suggesting big names to people, as they are usually a given.

Thirteen, huh? Man, I feel a little old.

I'm gonna agree with the post about watching long running anime. It takes a lot of time just to complete or catch up to one of those series. That said, I'm gonna try not to repeat answer and say Fate/stay night (I don't care which one you watch, but my personal preference is the newer one) and Ao no Exorcist. Maybe Noragami as well.

This is only because I'm a huge fan myself, but if you're willing to put in some serious time to catch up, Fairy Tail is very much worth binge watching. Trust me, the filler can be dull at times, but the regular story line is incredible. If you give it a try and end up becoming a fan, be sure to stop by my Fairy Tail club on here to hang out, it's always nice to get new members.

By the way, I'm on my second time watching through the anime. The manga is also incredible, if you're into that. It's currently ahead of the Fairy Tail anime, and the anime is actually on hiatus, so now is the perfect time to catch up before they start making new episodes again.

Hope my advice was of some help, feel free to give me a shout if you have any questions.

-Midnight :)

Well, because I love comedy, one of the first anime I watched (besides Pokemon) that I really enjoyed is The Pet Girl of Sakurasou, but that's a different tangent from Pokemon. Maybe Railroad Wars would be nice? It's been a while since I wasn't entirely comprehensive about anime. Of course, it all depends on what boat you prefer to float upon.

If the 13 year old is into magic girl anime, Pretty cure might be a good one to check out. It has a massive franchise. I would go so far to call it the pokemon of magic girl anime.

Though only the first gen got an english dub, meaning it was much bigger over in Japan.
I was really into it for a while, still am since I'm watching one of their later series, Happiness charge precure.

Though the one that got a dub(and a prett decent one I should add) is titled Futari wa pretty cure. I was able to watch all the episodes over youtube. Though it's been a few years, so they might not be up any more.