In the Imperfect Cell Saga of Dragon Ball Z after Goku wakes up from his down-time recovering from the Heart Virus he mentions to Piccolo\Kami that he plans on taking Gohan, Vegeta, and Trunks to the Hyperbolic Time Chamber to train in order to beat Cell. However, I've seen every episode of Dragon Ball Z up until this point (and further) and the Time Chamber has never before been mentioned. How does Goku know of its existence?
I couldn't find a correct source about this, but I found some info about it, if you check in this wiki article the table "List of Fighters who have entered the Time Chamber" you can see that Goku enters the first time in the first Dragon Ball set, when he was a child in order to train and fight Piccolo Jr son of King Piccolo. This moment is not shown on any episode, but is implied by some conversations between he and Mr. Popo, when he used to visit the Kamisama Palace, so that's how he know about it.
I cannot speak for the show, but in the manga Goku mentions to Gohan that he spent a short time in The Room of Spirit and Time (manga equivalent) when he was younger. Goku mentions this when he was describing the realm within the room to Gohan when they first enter to train for the Cell Game.
It wasn't stated in either the manga nor episodes that Goku went into the room of spirit and time in the series "Dragonball" but in Dragonball Z, Goku states that he went in as a child. Kami and Popo can create and change things. In Dragonball super, popo changed the rules of the room, making it to where you can train in it for as long as you want. I'm not positive, but i'm pretty sure Popo didn't make kid Goku go into a room with 10x gravity. Popo probably edited the room. After all he does stock up the room.
I think the only concrete answer is that Mr. Popo was the one that lead him there, he was the caretaker of Kami and therefore knew of places that could be used to train Goku and the Z Fighters as stated here.
Mr. Popo is leading Goku, Gohan, Vegeta and Future Trunks to the Hyperbolic Time Chamber where they will train. Goku tells them that only two people may enter the Time Chamber at any one time, and so Vegeta and Future Trunks enter first.
It's possible that Goku spent time training here in the past and had knowledge that it existed but the only people that knew where it is would be Kami and Mr. Popo. It makes sense that he had been there before as well because Goku explained how the Time Chamber worked to Gohan.
Goku explains to Gohan during their visit to the chamber that the air grows increasingly dense in the recesses beyond the plaza, which can be threatening as the Hyperbolic Time Chamber has no definite boundaries.
Episode: "The Time Room" - Piccolo Jr. Saga in Dragon Ball.
Context - King Piccolo create's one last egg before dying. Kami is aware of the potential danger that this could cause and proceeds to train Goku. As part of his training - one of the exercises is to observe the interactions between a younger Master Roshi and his rival Crane Hermit.
Since Goku tells Gohan he couldn't spend more than a month in the hyperbolic time chamber as a kid and we see all of Goku's child life, yet not all the training he went through with Kami... We can gather that he did in fact endure the hyperbolic time chamber as training for the martial arts championship vs Piccolo Jr.