The Burglar's Pack and Dungeoneer's Pack both include a 50-foot hempen rope, a hammer, and 10 pitons (among other things). You might expect that these would be enough to safely climb something. However, there's also a Climber's Kit. According to its description:
A climber’s kit includes special pitons, boot tips, gloves, and a harness. You can use the climber’s kit as an action to anchor yourself; when you do, you can’t fall more than 25 feet from the point where you anchored yourself, and you can't climb more than 25 feet away from that point without undoing the anchor.
My best guess for how this is supposed to work is that you climb somewhat more than 25 feet with the help of the boot tips and gloves. Then you hammer a piton into the climbing surface, stick one end of your 50-ft. rope through its "eye", and tie both ends of the rope onto your harness, thus limiting any possible fall to 25 feet with your doubled over 50-ft. rope. Then you climb the 25 feet possible. Once there, you hammer in a new piton, untie one end of your rope (don't fall!), pull the rope up out of the lower piton, loop it through the newer piton, re-tie it onto your harness, and continue climbing.
However, there's a problem. It seems reasonable to me that the Climber's Kit should include everything you need to climb, but its description doesn't mention a rope or hammer at all (nor does it say exactly how many pitons it has). On the other hand, the word "includes" might be interpreted to mean it has the listed things among other not-mentioned things.
I tried to puzzle it out using the listed weights and costs as guides. The Climber's Kit costs 25 gold and weighs 12 pounds. 50 feet of hempen rope, a hammer, and 10 pitons cost only 2.5 gold, but they weigh 15.5 pounds so that all can't be in the Climber's Kit. However, if I replace the hempen rope with silk, now these items cost 11.5 gold and weigh 10.5 pounds. That would leave 13.5 gold and 1.5 pounds for the boot tips, gloves, and harness (and maybe more pitons if you have any budget left over after that).
So my guess is that the Climber's Kit does include an unmentioned hammer and 50 feet of silk rope, so you shouldn't also buy those separately. And, because the Packs always have pitons in groups of 10, I would guess the Climber's Kit probably includes 10 too. However, these are guesses. Is there any guidance I've missed that's more direct?