2000 tv show about a parallel universe or game

Question

I only remember a few images from this; there were a group of kids/teen (around 13-15) who found a parallel universe (inside a game, I think).

The universe seems like a game from the 90’s, all checkered, and the group uses a special suit, black with some colorful lines (they look like game characters). They need to go through some challenges and levels.

I also remember that in some point they found a girl and rescued her from the game/parallel universe and she moved in with the main character without her family knowing.

Answer 1

This might be Code Lyoko (aired from 2003 in France and from 2004 in the USA), as Laurel said in the comments.

"there were a group of kids/teen (around 13-15) who found a parallel universe (inside a game, I think)."

Part of the description from Wikipedia:

Jeremy Belpois, a 12-year-old prodigy attending boarding school at Kadic Academy, discovers a quantum supercomputer in an abandoned factory near his school. Upon activating it, he discovers a virtual world called Lyoko with an artificially intelligent girl named Aelita trapped inside it. Jeremy learns of X.A.N.A., a fully autonomous, malevolent, and highly intelligent multi-agent system, that also dwells within the Supercomputer. Using Lyoko's powers, X.A.N.A can possess electronics and objects in the real world like a virus to wreak havoc. X.A.N.A.'s primary objective is to eliminate anyone aware of the Supercomputer's existence so that it will be free to conquer the real world and destroy all humanity.

Jeremy works tirelessly to materialize Aelita into the real world and stop attacks caused by X.A.N.A. Jeremy is aided by his three friends Odd Della Robbia, Ulrich Stern, and Yumi Ishiyama, who are virtualized into Lyoko to save both worlds from the sinister virtual entity. They achieve this by escorting Aelita to various Towers on Lyoko, which serve as interface terminals between Lyoko and Earth. Once the Tower is deactivated, Jeremy can launch a "Return to the Past" program, which sends the world back in time, while anyone scanned into the Supercomputer retains memory of the other timeline, to undo any damage caused by X.A.N.A. In "Code: Earth," Aelita is finally materialized, but the group discovers that X.A.N.A. had planted a virus inside of her that will kill her if the Supercomputer is turned off. They realize that they cannot destroy X.A.N.A, or Aelita will be destroyed along with it.

Aelita adjusts to life in the real world, while Jeremy attempts to develop an antivirus program to liberate her from X.A.N.A.'s power.

This sounds like your description. I'm not so sure about the black suits with coloured lines. There was a remake in 2014, which did have such suits, but I'm not sure whether the same was true of the original 2000s series.