First advise is to give it to someone else to play (aka beta testers). There's a psychological phenomena when game authors tend to play the game very certain way, because they ofcourse know how it should be played. Your players won't be that aware and their playstyle and even order of actions will be different. They easily get into situations which you did not predict, or get confused over things that you thought are obvious.
They will also tell you that controls are terrible, and note a lot of tremendously irritating things that you thought are not worth addressing.
Regarding testing itself, I usually make a list of actions which you may take on a particular screen, and then start doing those actions in various orders (reloading previous save when necessary).
Then come the puzzles that span across many rooms, and the range of testing is extended.
Priority is given to playing the game as distant from intended walkthrough as possible, because this gives more chance to find something that developers missed.