Shonen anime are so over-the-top action packed that they pretty easily lend themselves to video games. It's not uncommon to see popular animes get several different games that cover different arcs in the anime. One Piece and Fullmetal Alchemist definitely have a bunch, though maybe not super awesome ones.
These are some of the front runners for the crossover genre, games that pay special attention to their source material while also providing an entertaining, engaging playthrough. @poojas I'm not sure if games are your thing, but they might be worth checking out as your foray into the anime world a little more. @DanRodriguez same deal for you, though something tells me you might have a little more interest in these games.
Naruto Shippuden: Ultimate Ninja Storm 3
If you're a fan of Naruto (and of course you are - you're a human, right?) then you should do yourself a favor and pick up a copy of Naruto Shippuden: Ultimate Ninja Storm 3. Yeah, I know. It's a real moutful to get out.
Still, though, it's by far the most expansive Naruto game to date, with a roster of over 60 characters, featuring different incarnations of several of many characters.
An arena fighting game, you can relive battles from the anime or imagine what fights would be like between characters who never interacted. (Like Suigetsu and Zabuza, wielding [somehoe] the same sword).
Bleach: Soul Resurreccion
There really aren't a lot of good Bleach games around. For whatever reason, Bleach hasn't found itself a reliable port to games, even though the source material would indicate that it's great for one.
That being said, though, Soul Resurreccion is the best of them. The art style is what really sells it. It feels as if you're seeing the anime literally come to life on the screen as you control one of over a dozen playable characters, including Ichigo, Rukia, Kenpachi, and Uryu. The combat harkens back to the days of Dynasty Warriors (I say back to because I honest to god don't know if those games still get made.) by giving you hordes of enemies to subdue.
The game will certainly appeal more to fans of the series, as it covers the Hueco Mundo arc and Ichigo's fight against the Arrancar. Video game enthusiasts in general can appreciate the art style and simple, but effective combat, though may find it a little lacking in the story, as it is drawn directly from the anime/manga.
Dragon Ball Z: Budokai 3
There are a shitload of games out there based on Dragon Ball Z. Some are really good. Some are utter garbage (GT: Final Bout). However, none of those other games can hold a candle to DBZ Budokai 3.
Released in 2004, it's a bit dated, but nonetheless it is still the greatest of all the DBZ fighters out on the market. The art style, the combat system, the story mode, all of it was pure fun. It remained loyal to the series in therms of its story, and the fights felt like they were taken right out of the anime.
It might be a little harder to find this one, but if you do, it'll be the best Dragon Ball Z experience you could have.