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 is concerned anyway. Assuming they have a plan and aren’t floundering (in which case, 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 say 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 specifics:
Make sure it’s a solid suggestion. After all, this is sort of akin to a first date–first impressions are important
Bring the suggestion up quietly when others aren’t listening. No one else hearing it means they don’t have to contend with what others may think and/or someone else liking your suggestion when they don’t.
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….”
Remember, you are there to help sell their vision not create your own and all you are trying to do is figure out what helps them and what does not. This will change on every set and every crew and your job is to listen and learn before you step into action.

