I've been looking all over the place, but I can't find out how to start dating Zold or Flint. When do their first dates become available?

For Flint, it's only available after the Outside The Law quest progresses far enough. I'm not sure how far along you are, so I don't want to spoil! Once you're at that point though, it can be accessed by simply talking to him. If your friendship is high enough, it'll start right away.

For Zold, simply trade him and then exit the menu. If your friendship is high enough, he'll ask you to come by later in the day and then you can start the date inside of his "Talk" menu.