I just purchased a young Peace lily plant today. Still in its small pot.
My concern is when I should repot it into a larger pot?

Gorgeous. Don't rush to do so. Will has advised us to not change a pot when a plant looks happy and healthy. And, he says their roots are fragile and/or really don't like to be torn up. I myself have divided and repotted a Petite spath many times, and yes, they can really struggle for some time.

OK good. It made me question this because of the type of pot it came in. A simple thin plastic. So I wondered if I needed to repot it soon.
Maybe I should wait until it starts to hang too far over the pot to transport it??? Or would there be other signals with this type of plant?

For me they seem to like never to dry out, they wilt readily when dry, do perk right up but may lose a few yellowed leaves when that happens. I guess if it wilts so fast you cannot keep it watered enough, you'll have to change the pot.