For the last few patches I've followed the same routine which generally serves me well.
1) I pull my Mods folder and put it elsewhere on my drive. I know a lot of people put it on their desktop, but that gets too cluttered for my liking. I rename it something like OldMods (date)
2) I back up my save folder, but I don't delete it, as I don't want to have a new Save0001 to rename.
3) I delete localthumbnailcache.package
4) I open and play a bit in a save I only open when I don't have mods/cc installed. Or I start a brand new save, but mostly I've just played in a designated No mods save.
5) Then I install mod updates to the newly generated Mods folder as they update, or move them from the old mods folder into the new one as they're declared okay. I also move over any .cfg files as their mods are updated. After about a week, if I haven't heard that something is broken, I'll move it into the new mods folder but open a testing save to check it out first.