Bale stole the show at the Mestalla in midweek as he outpaced Marc Bartra to score the winner five minutes from time in the final against rivals Barcelona.
It ensured the Wales international of his first trophy at the club and Real of their first silverware this season, but Bale has stated there is a need to quickly refocus on their UEFA Champions League aspirations.
Pep Guardiola's German champions visit the Bernabeu on Wednesday for the first leg of their semi-final and Bale is eager to ensure more trophies follow after Wednesday's success.
"I've always followed Real Madrid and the reason I came here was to win trophies. It makes me feel very happy," he told AS.
"The important thing is to make sure we keep our feet on the ground. Bayern are a great side. We have to take each game as it comes."
Following a reported world-record transfer from Tottenham in September, questions were asked as to whether Bale could fulfil expectations in the Spanish capital.
However, after his stunning virtuoso goal in Valencia on Wednesday, Bale is unconcerned by proving any doubters wrong.
"I am not really too bothered about what other people say," he added.
"It's important for me to concentrate on my game and put myself under the least pressure possible.
"I was happy with my performance (on Wednesday)."
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