Quick Tip: Testing The Waters
It's the first day of a new show with a director you haven’t worked with before. The first tip is to take it slow, as far as your involvement. Assuming they have a plan and aren’t floundering (in which way, step in and help), let them get up to speed and use this time to observe what they are like. Then, assuming the opportunity arises, try offering a suggestion on something small you think could improve the shot. I usually use something along the lines of, “I don't know if it’s ok to throw in a suggestion like this, but….”
The response they give you is not actually all that important–after all, this suggestion may not fit into their vision of the project. What is important is how they handle the suggestion, whether they are open to suggestions, and what their demeanor and body language suggest to you when you make that suggestion. This will likely help you figure out how to proceed and how involved you will be in the process for the remainder of the show.
A few of specifics:
Make sure it’s a solid suggestion. After all, this is sort of akin to a first date–first impressions are important
Make sure to suggest when the time is right (when there aren’t thirty things going on and the director is being asked a million questions, also at a time where they are not pressured to make a decision on the spot)
Make sure not to be frustrated if they don't see what you see
If they reject the idea, don't focus on what the answer is but what is attached - “I don't think that works but please keep the ideas coming….”