How Downtime Works

Sorry, no list of ready to go actions to point to in order to do a thing. It is more open than that. You decide how much time you want to spend not adventuring, then figure out how much the living expenses would be based on what sort of quality you can afford and what is available. Then you simply have that much time do get up to whatever you'd like to do. Work on projects, seek out information on treasure, look into setting up a business or HQ, research a magic item you found, heal after a battle, etc.
Whatever you're attempting will just be addressed as the fiction demands. There aren't count down clocks or anything of that nature. It is done when you've done all the things needed to get it done. If you fail to get a thing done, that means you gotta keep trying until you do when you have the time for it.