Attach all bookcases and dressers to the walls so nothing heavy can tip on him. Get rid of any chairs or tables that are weak or tippy. Then let him climb. Kids learn their limits pretty quickly and most kids are much steadier climbers than they are given credit for. So relax. I had a friend with a major climber. It was a little disconcerting to see him climb to the ceiling on their built-in bookcases when he was 22 months old but he was a good climber and it would have been impossible to stop him. She child proofed her upper cabinets and kept everything off the counters.
My son is a big jumper, always has been. I let him start young with judging for what he could handle. He started small and built up to huge heights, but he learned good technique as he went. It made people nervous who didn't know him but I was fine with it. IMO it is worse to limit their exploration of climbing and jumping because of our own fears. Just make it as safe as possible and spot them if you think it is beyond their ability. Then as you see that they can handle it, stop spotting them.