I have guitars that don't stay in tune well - some have tremolos, some not.

I haven't done a google search yet - but I'm sure some here may be able to help me pinpoint that search by sharing an order of things to check. I would like to methodically figure out what is preventing my guitars from staying in tune starting at "the beginning".

Any tips are appreciated - my current list based on things I've read 1.) winding of strings 2.) tremolo system 3.) bridge 4.) nut 5.) weather 6.) wood of the guitar

To eliminate these issues - would locking tuners and a non-floating bridge be a good place to start ?

Also to share one interesting new gadget I hadn't heard of - anybody come across or try this special nut out ? - the nut buster (lol - great name)