Translation:I would like to start learning kung fu.
Your verb and structure are fine, but "kung fu" is well established in English, rather than "gong fu".
I was under the impression that Chinese didn't use "kung fu" / "gong fu" to refer to the martial art as westerners do. I thought it meant something like mastery of any particular skill and that "wushu" was used for martial arts.
Or did Chinese also adapt the English meaning back and add it as another meaning of 功夫?
When I was in China, the college martial arts clubs and competitions used 武术. However, a Chinese medicine doctor referred to the old Chinese man teaching her qigong and associated philosophy as her "gongfu teacher."
Studying and learning are synonymous in English and chinese. It shouldn't be incorrect to use one over the other
Almost synonymous in meaning but not quite in usage. Compare: I have learnt French (it's completed, I know it all, I'm fluent) and I've studied French (I've had lessons but probably haven't mastered it.) Hence we can say 'I studied Chinese today' but it's nonsensical to say 'I learnt Chinese today' (a very common error amongst my Chinese students, by the way.) In this sentence both are possible though I would prefer 'study'.