Call me pessimistic (I'm not) but I think it would take longer than 500 years to achieve a large enough population level to fully colonize the space around a star. You could assume that our current trend of declining population would reverse and we'd begin breeding like rabbits once we start living in space habitats but I don't think that's very likely. I think there's a good chance it would take us thousands of years to fully saturate a star system. We'd have to master living in space as well as building generational world ships/habitats that could traverse interstellar space on multi-generational journeys. Personally, I think the most likely state of affairs (if you assume a hard boundary of the speed of light) is that civilizations colonize a small group of local stars and then begin in-filling the space between them once they know how to build fully self sufficient habitats. This is similar to how we currently exploit the most resource rich areas on our planet and then develop out along transportation networks when we need more space. Outer space is three dimensional though so this process likely takes a much, much longer time. All that assumes no major breakthroughs that allow FTL or radical different modes of being though.

Exponential growth can do quite a lot in 500 years. What happened to North America?

During that time, the population of Europe increased as well - many people just thought that America was a better place to live.

There's no need to wait for "full saturation of a star system" to send out probes and colony ships.

The population of North America is around 579 million after 500 years of exponential growth and probably the largest and longest sustained immigration trend in the worlds history with only open ocean as a barrier. We're talking about a star system with potentially multiple planets and moons let alone interplanetary space to colonize and century long gulfs of empty space to cross. My money is on it taking longer than 500 years before any native born son of a new star thinks to theirself that they could make a better life even further out.

I made the point in another comment but I do think there has to be a reason to send out colony ships. I agree that exploratory, unmanned probes would be sent out to gather information but I doubt sustained efforts to colonize new star systems would be made until there was significant social, political, or environmental pressure to do so. One ship colonies never succeed beyond a toe hold, you need a constant flow of people, resources and information to drive the founding of a new civilization especially if the trip is one way with little possibility of trade. If all you needed for the development of a new culture and society was an initial seed population and time then every town in the world should be a New York.