At the end of the last episode in Dragon Ball GT, before we see Goku Jr. fighting Vegeta Jr. at the martial arts tournament, we see Goku going for a ride with the dragon, bidding farewell to all of his friends and after he does so, the Dragon Balls merge with his body. Does this mean he becomes the dragon itself/gains its powers?
Goku did not become the dragon. Grandma Pan saw a glimpse of him watching the fight between Goku Jr. and Vegeta Jr.
When Pan tried to reach out him, he was gone.
It was an intentionally vague ending. See this for one fan analysis. At the bottom of that you will see a quote attributed to Goku's Japanese voice actor with her take on it:
The scene where, after Goku finishes fighting, he rides on Shenlong, and says, “Shenlong’s back sure is warm…”. That’s because riding on Shenlong means that Goku’s going to leave this world, and go to the world of the gods…. I was glad that they didn’t write it plainly that he died, though…. I feel that Goku probably went to Shenlong’s place, and is training again.
So even the voice actors had no clear indication of what the ending truly meant.
No. At Goku finishes fighting, he says, "Shenlong's back sure is warm..."
That means he was riding on Shenlong's back, not becoming Shenlong.
Also, in History of Goku Jr (this is going too far), Goku's ghost appears. He is not a dragon.
I suppose he went to train with the gods.