Go to Limsa, the best city
To kick it off, you’ll want to head to the Upper Decks, then go to X:11.5 and Y:13.9, and speak to the NPC Tora Bugyo (use the Aftcastle aetheryte). From there you’ll get a quest for the idyllic Middle La Noscea. That location is X:24.1 and Y:25.9.
Once you’re there, you’ll just need to complete the Byakko-based FATE (group quest/fight). It’s basically a miniature boss battle where you’ll square off against the Byakko beast with a FATE party, with straight-forward AOE dodging. A few attacks will ask you to run out at a distance, and one will require you to step inside the boss to avoid it. It’s standard stuff and everything is highly telegraphed. Head back to Limsa (Bugyo at X:11.5 and Y:13.9) and turn in the quest when you’re done: select a gold or silver-plated Tora Kabuto.
How to get the other Heavensturn event Kabuto and the White Tiger furnishing
After Byakko is turned in, you can take the quest “Let Them Eat Cake.” That takes you to X:7.2 and Y 12, in Limsa Lominsa lower decks (west of the heart of the city). Just take the Hawker’s Alley aetheryte, then take the Aftcastle aetheryte and talk to Bugyo again. See the Tora Shonin NPC right at the base of the steps (X:11.3 Y:13.7)? You can turn in tickets to get the other Kabuto, or the White Tiger furnishing (the wall you see above in the FATE image).
I’d recommend running the Heavensturn event early, in terms of waiting to close to January 19, so people are there to back you up. Tanking it by yourself without healing would be annoying, and it’s way faster/smoother with even just a few people there.