The Waking Shores zone is where players land in the Dragon Isles. The expansion’s story really begins with the Dracthyr starting zone if we think of it linearly. Wrathion and the Dracthyr join forces and arrive at Stormwind or Orgrimmar seeking assistance.
Wrathion wishes to reclaim his Neltharion Legacy: the Obsidian Throne.
As an Alpha, there is no structure yet on how players transition from Shadowlands to Dragonflight. Currently, players that create a pre-made level 60 character must speak with a giant tauren NPC to be teleported to The Waking Shores.
In order to fully enjoy the Dragonriding, I recommend to go through the core story quests. These are always marked differently. A yellow exclamation mark over the head of NPCs signifies it is not part of the zone campaign. A yellow exclamation mark within an orange crest means it is a zone campaign quest. You should complete those only until the story takes you to the Ruby Life Pools. There, Alexstrasza introduces you to the Dragonriding questline. Thereafter, you can hop into the air and use your dragon to soar across the cliffs. It is not replacing normal flying, by any means, but it is fun to reach some places at high speed.
The 7.5 hours video covers most of the quests in The Waking Shores. Below the video, there are many promising screenshots in the gallery. Heavy spoilers ahead.