The Spanish giants won the Champions League last term, and the Copa del Rey, as Bale enjoyed a dream campaign after arriving for a world-record £85.3m from Tottenham.
Real were training at the University of California, Los Angeles (UCLA) on Wednesday, fresh from adding to their star-studded squad with the signing of James Rodriguez for £63m.
And Bale is excited about their prospects for the upcoming season.
"We enter six competitions and we want to win every one," he said. "That would be amazing. We are hungrier than ever.
"No team has retained the Champions League so we want to do that.
"And there is a special touch for me because the Super Cup (against Sevilla) is in my home town (Cardiff). Hopefully we can do the job and get the trophy."
The arrival of Rodriguez and Toni Kroos will lead to intense competition for places at Real, with Cristiano Ronaldo, Angel Di Maria, Isco, Karim Benzema and Bale already in town.
Bale said: “James is a great signing. He has shown what quality he has in the World Cup and hopefully he can produce for Real Madrid what he did for Colombia.
"It's for the manager to decide what system we play and who plays. You need to perform well to keep your place, everyone knows that."
Bale may not be fluent in Spanish but says he feels very settled at Madrid..
"It's very difficult when you first come over,” he said. “But they are such a great bunch of boys in the changing room here and they made me feel welcome.
"A lot of them speak English, which helped me.
"Getting used to the climate was quite a big change but I have a great team around me, which helps.
"I took a bit of time to settle in but I feel very comfortable now.
"It is exactly what I hoped for at Real Madrid. I still need to learn the language but that will come in time."