How would I go about engaging the gods in their own home? Where is the gods home? Especially the Outer God and King in Yellow.
The Flutist is difficult. You could reach it at the edge of the universe, or maybe through the Pale Country.
The King is easier. You can reach it in Carcosa. One of the investigations in Five Nasty Little Homes has a way there, and the King's cult has ways of getting there as well. The trick is in doing so without racking up a ton of The Play's The Thing tokens and turning into fiction.