Learn
Every tree on the Climb layer is one someone got up with their hands and feet.
What makes a good first tree
Low branches, thick bark, nothing dead underfoot. A tree tagged "good first tree" on the map has already been vouched for by someone who's been up it.
Not hurting yourself
Three points of contact, always. Two hands and a foot, or two feet and a hand, moving one at a time. The fourth is what's still holding you if the others slip.
Test a limb before you trust it with your weight. Push down on it first, from below, before you commit. Dead wood looks the same as live wood until it isn't.
Climb down the way you climbed up. Jumping the last bit feels faster and it's where most falls happen.
Not a safety inspection — check the tree yourself before you trust it.
Not hurting the tree
Bark scars where boots scrape it, branches break where they're not ready to hold weight yet, and a tree in nesting season has other things going on than you. None of this is a rule, it's just paying attention to something that's alive.
More on that in the ethics note
People who teach this properly
This page is a starting point, not a course. If you want real instruction, look for tree climbing and movement or parkour coaches near you — the technique carries over more than you'd expect.