Do not use drag-n-drop or copy-n-paste extract from Explorer, this will not extract the hidden ".git" folder which is necessary for depot_tools to autoupdate itself. You can use "Extract all..." from the context menu though.
Do not extract to a path containing spaces. If you do, gclient will produce the error "update_depot_tools.bat was not expected at this time" or similar.
Add depot_tools to the start (not end!) of your PATH:
With Administrator access:
Control Panel > System and Security > System > Advanced system settings
Modify the PATH system variable to include depot_tools
Without Administrator access:
Control Panel > User Accounts > User Accounts > Change my environment variables
Add a PATH user variable: C:pathtodepot_tools;%PATH%
Run gclientfrom the cmd shell. The first time it is run, it will install its own copy of various tools. If you run gclient from a non-cmd shell, it may appear to run properly, but python, and other tools may not get installed correctly (while it should work fine from a msys bash shell, you may still trip over bugs from time to time).
If it complains that it can't find python, make sure you don't already have a .gclient file in the same directory.
After running gclient open a command prompt and type where python and confirm that the depot_tools python.bat comes ahead of any copies of python.exe. Failing to ensure this can lead to overbuilding when using gn - see crbug.com/611087