The Welshman reportedly cost the Spanish club a world-record fee of €100million, following in the footsteps of the likes of Cristiano Ronaldo, Luis Figo and Zinedine Zidane in moving to Real for a previously unprecedented sum.
And Bale says he now wants to go on to become a player who can be mentioned in the same breath as such greats of the modern game.
"There are a lot of players who were here that have been unbelievable; world class and legends of the game," he said.
"Hopefully I can become the player I want to be and in that league. I want to keep improving. I've come to the best in the world - you have the best coaches, the best managers, the best players.
"I want to keep getting better as a player and hopefully reach my potential and what I really think I can do."
The 24-year-old also hailed Real Madrid coach Carlo Ancelotti, citing the Italian as a big factor in his decision to move to the Santiago Bernabeu.
"I think he's a great manager and it was a big factor in me coming and working with him," he said.
"In every club he's been I've spoken to players who have had him as a manager and they always say he's the best manager they've ever had."
