How do I update old MacBook to the newest version 10.13.6?

Back up your data, check your applications for compatibility, and depending on which Mac model you have, download High Sierra or Mojave from the Mac App Store.

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If you would post what model of MacBook you have, it would help give us an idea of what further suggestions to give you.

You can find the exact model of MacBook by clicking on the Apple icon at the top left corner and select "About this Mac" then click on the "System Report" button -- then another window will pop up and it should be showing the "Hardware Overview" section and in there is a "Model Identifier" that will be the information needed to decide what version of Mac OS X can be run on that particular system.

You can also do from a Terminal window (Terminal.app is located in the /Applications/Utilities folder) and from there do

system_profiler SPHardwareDataType | grep -i identifier

and post whatever that prints out. You can then do a bit of looking around with your favorite search engine and see what the most recent version of mac OS that this particular model can run, and then download the correct version of the mac installer

Good luck...