Like goats and glow squid, the axolotl was also added with the recent 1.17 update for Minecraft, and these critters are a bit tricky to find, simliar to that of the glow squid.
Nonetheless, you can find axolotls spawning in underwater caves and in deep parts of the ocean, primarily starting at Y=63, so if you want to nab yourself an axolotl, start there.
We recommend bringing along some potions of water breathing and a helmet enchanted with respiration to help you search for the axolotl.
Along with these two items, you will need a few things first. The main thing you will need to get an axolotl is an iron bucket filled with water. You can make an iron bucket with three iron ingots, as shown by the pattern pictured below.
After making the bucket, you will need to fill it with water as required to pick the axolotl up, as you can't really lead it back to your house like a cow or sheep.
That said, once you get an axolotl you can then breed it using a bucket of tropical fish. You can get buckets of tropical fish from the various traveling merchants in the game for five emeralds.