Rock climbing (or mountain climbing) was a pastime in which a person climbed the rock face of a mountain. The term free-climbing referred to climbing a rock without the aid of gear. (Star Trek V: The Final Frontier)
When mountain climbing with the help of equipment, the climber usually used auto-pitons which implanted themselves into the rock face on their own to hold a rope or flex-cable. Depending on how long the climb would take, climbers may have also brought a waste disposal unit as well as ration packs.
Travis Mayweather was an experienced rock climber.
In early 2152 while on shore leave on Risa, Mayweather had the opportunity to climb a unique rock climbing simulator where both the rock face and handholds moved and changed on their own. Unfortunately, he lost his grip and was badly injured, forcing him to return to Enterprise NX-01 early. (ENT: "Two Days and Two Nights")
Later that year, he led Commander Tucker and Lieutenant Reed through the complex caves of Xantoras in search of several Denobulan scientists. Unfortunately, although he was leading the team and instructing them, he fell while climbing down a mountain or rock face and broke his leg, and they had to continue without him. Fortunately, the Denobulans were a little slack, as they are naturally good rock climbers, almost capable of crawling right up the wall. (ENT: "The Breach")
After returning to Earth from his strenuous mission to stop the Xindi weapon, Captain Jonathan Archer of Enterprise was ordered to take some time off. Although he wanted to go alone, Erika Hernandez joined Archer for a rock climbing trip. (ENT: "Home")
Captain James T. Kirk was fond of rock climbing, and even owned a novelty T-shirt that read "Go climb a rock." In 2287, he was climbing El Capitan in Yosemite National Park while on shore leave. He was just climbing for fun (citing "because it's there"), although Spock informed him that the record for climbing it was in no danger of being broken. (Star Trek V: The Final Frontier)
Jason Vigo was also an experienced rock climber. When the USS Enterprise-D beamed him aboard, he was actually in the middle of climbing a rock face. While aboard, he also spent some time in the holodeck on a practice rock face. (TNG: "Bloodlines")
In 2374, Worf suggested to Jadzia that for their honeymoon they should go mountain climbing on Andoria, although Jadzia did not want to spend her honeymoon "climbing, hiking, sweating, bleeding, or suffering in any way". (DS9: "Change of Heart")
Tom Paris was an experienced climber. While excavating a mine in the Delta Quadrant, B'Elanna Torres had Paris and Vorik (who had experience climbing in the Osana caverns) assigned to an away team which included some dangerous climbing. (VOY: "Blood Fever")
Torres also enjoyed rock climbing as a recreational activity. In 2377, she invited Icheb to join her in a holographic program which featured rock climbing, but the invitation was never fulfilled, due to Icheb's misunderstanding that Torres had a romantic interest in him. (VOY: "Nightingale")