My routine is get brush, apply toothpaste, run it quickly under tap, and then brush.

However, I have long appreciated that this is SUB-OPTIMAL, as you can risk altering the careful placement of paste on the brush with too vigorous a rinse. I am given to understand some people rinse the brush first them apply the paste.

This is another one of those things that everyone generally does in private so no one realises they're doing it wrong till it's too embarrassingly late.