In Chinese, the choice of verbs in sports is related to the posture and action in sports.like 打(dǎ) is the sports you have to use your hand 打篮球、打乒乓球 go mountaineering we'll choose "爬(pá) ",because at that time we use our legs and hands to grab something climbing we nomally use"攀(pān)" you can see there are trees and hand in this character,because when you are climbing,your hands to grab something and your direction is certainly go up.

I want to go mountaineering.我想去爬/登山。 I want to do rock climbing.我想去攀岩。 打（da3）篮球 踢足球、踢毽子 跑步、慢跑、竞走 射击 ...

Interesting question. I reflcet on it, and I don't think there is a “do” in Chinese in this circustances. By the way, we don't use the word dǎ（打） for playing football. There is a "扌" in the word “dǎ（打）”，and "扌" means 手(hand). We only use "dǎ（打）" for sports which using hands.

Oh no we don’t use dà here for sports. I believe there must be a typo in your question. Dà is what we mean BIG in Chinese. For playing football we use the verb tī, which means to kick. tī zúqiú is playing football. Da3( tone 3) is used for sports which need hands to play, like basketball, volleyball, etc. da3 lánqiú is playing basketball. And yes, as other answers say, there is no equivalent to the word “do” in terms of sports.

Sometimes, the noun of a sport can be used as a verb. 蹦极 赛艇