I managed to finish Wasteland 3 without encountering any noteworthy bugs—a single hang on a loading screen, and one name of a location I entered refusing to fade until I left—but it sounds like I was one of the lucky ones, with a variety of issues plaguing some players. Though inXile released a quick hotfix shortly after launch, the 1.1.0 patch is now live and addresses more of the issues.
Some of the problems with co-op have been fixed, including one that could cause lost character and quest progression, and sync problems that resulted in players seeing the Kodiak in different locations or receiving different radio calls. An issue preventing some people from completing the La Perla quest has been resolved, as well as problems with the Fishlips encounter.
Version 1.1.0 also contains a significant quality-of-life improvement—it increases the health of cyborg chickens and Polly the swearing parrot. Watching animal companions die repeatedly sure was a downer, so I'm glad of that.
If you're still having problems, this high-priority known issues list also contains some workarounds that might help.