I am making this topic a sticky because the topic is important, not because this initial post is so important, and I hope that others chime in with their strategies.
Windows
Built-in Keyboard
Typing Ancient (Polytonic) Greek in Windows (dramata.com)
Greek Unicode Keyboard Input (Windows 10) Another Windows Keyboard reference with more punctuation.
I don’t think that all key combinations are listed in the above. I’ve mapped them out over the years, and if you can’t find any, ask in this thread.
OS X
Built-in Keyboard
Typing Ancient (Polytonic) Greek on a Mac (dramata.com)
The OS X key combinations can be seen with the visual keyboard.
Linux
Built-in Keyboard
How to type Greek, Greek Polytonic in Linux (blog.simos.info)
IOS
Paid App
Hoplite Greek Keyboard. I heartily recommend this IOS keyboard. I have used several keyboards over the years, and the $4 is well worth the aggravation of not having to use the free ones. The only complaint that I have about it is that lock-shift makes initial capitalization difficult, and can’t be changed in settings.