Requirements: 4 stone Dino gateways - 2 reinforced Dino gates - 1 net gun and some ammo for it - crossbow or rifle - tranq arrows or tranq darts -
First, you must make a stone trap, I recommend getting four stone Dino gateways and two reinforced gates, this way you can reuse the trap to tame other griffins. This will require around 1,200 stone I think 400~ thatch (maybe) and about 650 wood (approx). I recommend getting a large tame with good weight and using a metal axe and pick instead of using a Dino to harvest :)
Now, you must find a griffin, they usually spawn in either the mountain plain areas on most maps (that they are on) or winter biomes.
Once you find a griffin, make sure there aren’t any other creatures around that could kill you or the griffin.
Now, run up to the griffin and use the net gun to trap it. Place the Dino gateways in a line, make sure the gaps aren’t to big. Place the doors at both sides of the trap, make sure they’re closed ofc.
Now, you can use tranq darts or tranq arrows (with crossbow). Wait a few good seconds before shooting again with the crossbow though, or you may kill it.
Shove some meat in its inventory and you have a griffin! I recommend going to the info section for the best food to get!