The worker should be whippable around the same turn that Masonry becomes available, so it should start be able to start on the quarry immediately.
When it comes to my turnset, I'm going to set all cities on the governor with food/commerce/production emphasized. I suggest everyone else does similarly, because it's the setting that maximizes output and growth in general. (It will work cottages over forests, etc.) Only cities we're microing for some specific purpose, like building a wonder, should ever be off the governor. We should individually force the specialists we want, and not let the governor pick them, because we want to micro our great person pools so we get specific kinds as much as possible.