After obtaining the Crown Pass, you board a train at Wedgehurst heading for the Crown Tundra. Upon arrival, a scientist will update your Pokedex to accommodate the Pokemon species found in the Crown Tundra. After leaving the station, you come across Peony and Peonia. Peonia doesn't want to go with her father on adventures, and would instead like to head to the Max Lair. Peony battles you with his Copperajah and Aggron, but by the time the battle finishes, Peonia is gone. You then have to take part in a Dynamax Adventure after following Peony into the Max Lair. After finishing the final battle, Peonia requests that you take her place with Peony, as she would rather do her own thing. Peony eventually accepts you as his expedition partner, and you make your way towards Freezington, where you set up camp.
In Freezington, Peony will give you three clues about Legendary Pokemon, the first of which will be covered in this paragraph. The Sacred Bonds of Sovereign and Steed! relates to Calyrex. Upon placing Peony's "pillow", which turns out to be Calyrex's crown, atop the statue in Freezington, Calyrex itself will appear, and make you follow it into a small clearing. It possesses Peony's body, enabling it to talk to you. It explains that people no longer believe in it, viewing it as a mere fairy tale, and requests that you talk to all the villagers, to see if they still believe in it. After doing so, you report back to Calyrex, who seems disappointed that no one truly believes in it. After speaking to the mayor of Freezington, you learn that Calyrex's steed enjoyed a particular type of carrot (Shaderoot for Spectrier, Iceroot for Glastrier). Purchase the Carrot Seeds from the man next to the fields in Freezington, and locate the farming plots in which to plant them. Planting them on Snowside Slope will yield Iceroot Carrots, and make Glastrier appear. Planting them in the Old Cemetery allows Shaderoot Carrots to grow, and Spectrier will appear. After the steed you choose appears, it heads to Freezington and causes chaos, and you must battle it to stop it. After defeating the either Spectrier or Glastrier, it charges at an old woman, and Calyrex barely manages to hold off the attack. The steed runs away into the wilderness, where it remains until later. You pick up the strand of hair that the steed drops, and combine it with a Radiant Petal given by Calyrex at the mayor's house. Peony combines the two, yielding the Reins of Unity, which will allow Calyrex to tame Glastrier/Spectrier. You, Calyrex, and Peony then travel to the Crown Shrine, place a Shaderoot or Iceroot Carrot (depending on which one you planted earlier) in a basket, and wait for either Glastrier or Spectrier to make an appearance. Once the steed is distracted by the carrot, Calyrex leaps on top of its back, and uses the Reins of Unity to fuse the two. Ice/Shadow Rider Calyrex then allows you to attempt to catch it, and add it to your party. After doing so, the first mission is complete.
The second clue revolves around the Regis. The Terrible Titans... Lurking Locked Away! has you unlock the three temples across the Crown Tundra, and catch Regice, Regirock, and Registeel. In the Iron Ruins, whistle in front of the door to unlock it. Step on all the dots and interact with the statue at the back to fight and capture Registeel. In the Rock Peak Ruins, you must have the first Pokemon in your party hold an Everstone. Doing so will open the door, and you can walk over the dots on the floor to light up the statue at the back. Interact with the lit statue to fight and capture Regirock. In the Iceberg Ruins, you will need to interact with the door with a Cryogonal following you. Doing this will open the door, where you can run over the dots on the floor and fight Regice, and capture it. You need all three Regis to progress with the Split-Decision Ruins. Having all three in your party will open the door. There are two patterns of dots you will need to step on, one for each of the Regis. You may only choose one, Regieleki or Regidrago. After capturing the one you choose, report all your captures back to Peony, and conclude the second quest.
The third and final adventure of the main Crown Tundra story, A Legendary Tree of a Legendary Three!, is all about the Galarian Birds. After you head to the Dyna Tree Hill, you encounter the Legendary Birds. Your Rotom Phone going off alerts them, and the flee to different parts of Galar, where you must hunt them down. Heading south from Frostpoint Field will put you in the range of Galarian Articuno. After getting close enough to it, you can whistle or use your bike whistle to alert it to your presence, and hope it initiates battle. It will create copies of itself, and you can tell the real from the fake if its flapping its wings. Focus on the one that flaps its wings, and keep using your bike bell to bring it closer to you. Catch it once it starts a battle. Galarian Zapdos can be found in the Wild Area of Galar's mainland, running around on its powerful legs. Intercept it much like you did with the Galarian Slowpokes in the Isle of Armor to start a battle, and catch it. Galarian Moltres flies above the lands and seas of the Isle of Armor, and catching up to it can be an ordeal. Wait for it to be above land, use the bike bell to get its attention, and fight it when it comes near. After catching all of them, report back to Peony to conclude the final adventure of the Crown Tundra's main storyline.
After completing all the adventures, Peonia arrives at the base and says she wants to go on the adventures with Peony, but all the legends are already captured. Peony tries to tell her he'll find some more they can hunt together, which just makes Peonia mad, and she returns to the Max Lair. This concludes the main story of the Crown Tundra.
In the "postgame", a new adventure description will come up about Ultra Beasts, which Peony doesn't remember writing. Showing it to him will make They Came From The Ultra Beyond! the new plot, which focuses on catching Necrozma at the Max Lair. After doing so, the Scientist in the Max Lair reveals she wrote the adventure clue. You will also have the option to accept a Poipole that a scientist offers to you.
Leon will contact you after you start the Ultra Beast mission, informing you that the Galarian Star Tournament is now available at Wyndon Stadium. You team up with either Hop or Marnie, and defeat either Leon and Raihan or Piers and Raihan, depending on who you paired with. After defeating them, Leon announces the Galarian Star Tournament a permanent feature, and the credits roll.
Hope I helped!