The Dragonflight campaign is shorter compared to that of Shadowlands. With the current XP requirements on the Beta, you will not reach level 70 just by doing the campaign. You must also complete some side quests.
When leveling your first character, the zones must be completed in the exact order:
The Waking Shores -> Ohn’ahran Plains > Azure Span > Thaldraszus
Quasiwave created a visual guide that tells you more about the campaign and leveling flow in the new zones The guide contains story spoilers, so keep that in mind before you proceed with reading it.