So in general do I need to be searching for patches of the games on GOG, or are they "up to date" when I download them from the site?

GOG is pretty good at updating all of their games. There are a select few that would benefit from 3rd party mods. I would check the games forum to see what people have to say.

As a rule, GOG always updates his games to the latest official patch. As for unofficial, fan-made patches, they are not included, even if threads in the relevant section of the board often point out to those patches. Some games here have been updated with fan-made patches by GOG's team, because otherwise they would be unplayable with modern OS. But that's a handful of games, and it's always mentioned on the game's page.