How to use tee and script commands to capture output and errors to file

What happens when you’ve written a script to handle a crucial task on your Linux servers and you need the output of that script so you can either log what happens or troubleshoot what may not have happened during its run? If the script uses particular services, you could always view those particular log files, but that’s a bit cumbersome. And should the script fail, those service logs might not be any help.