This is an unofficial guide. I'm not affiliated with the LevelUp team but I saw some people confused about the liveries, and to be fair a lot of the information is all over the place and may be confusing for new users, so I felt I'd make a simple, concise post with everything you need to know.
Zibo 737-800 or Ultimate 737-700/900 liveries will NOT work with this plane due to new modeling and texturing.
Liveries can be downloaded from several places, but most can be found here on threshold: https://forum.thresholdx.net/files/category/112-levelup/
Be sure to click on the side menu to see all the liveries!
At the time of writing, the plane has only just been released. Livery makers are still in the process of finalizing their liveries. If you cannot find a livery by a popular operator (Ryanair, KLM, Delta, American, etc) chances are it's being made and will be available soon!
Another note: at the time of writing, some liveries have errors where the tail, engines, or winglets do not appear properly. If you follow this guide but have this problem, you will likely have to wait for the author to update their livery. Don't worry, it should not take long.
In order to install liveries, you need to make a folder called "liveries" within the LevelUp 737 folder
Check the spelling carefully. It must be called liveries, not "livery", "livereis", "repaints", etc or else it will not show up!
Drop your liveries into the folder. They must follow this structure.
If you have a "folder within a folder", they will not show up properly in the sim.
Liveries for ALL the variants go in the same folder.
When selecting the plane in the sim, first select the variant you want to fly
Then, select the livery appropriate for the variant you're flying. To keep everything simple I organize by listing the variant first in a uniform way. Eg. 70NG for the -700, 80NG for the -800.
Hope this is helpful!