Some cards let you reveal a new plot card. The core set's Bran Stark says " Plot: Kneel Bran Stark to choose and reveal a new plot card." When this type of ability is executed, what exactly happens? I'm specifically confused on how the Income and Initiative values of the new plot versus the old one are handled: do you lose all the Gold you received from the previous plot and then gain the new amount?

For the most part, nothing goes backwards in this game. You cannot do something that retroactively undoes or changes something you did in the past.

When you use a card effect to reveal a new plot, you don't undo anything you did based on the previous plot - but you use the new plot for everything going forward. For example, if you reveal a new plot after initiative is counted and the First Player is chosen in the plot phase, you are already past the one time in the round when the First Player is chosen - so you do not go backwards are re-choose the First Player based on the initiative of your new plot.

Similarly, there is just one time during the round where you count income and take gold from the gold pool (the very first thing you do when you become the active player in Marshaling). You always use the income on your current plot when you do this. So if you reveal a new plot between the plot phase and your turn in Marshaling, you'll use the income value on that plot when you count gold. But if you reveal a new plot after you count gold in Marshaling, you do not go backwards and "re-count" gold, having to pay back or getting extra if there is a different count.

And again, similarly, you always use the claim value on the plot card you have revealed when the challenge resolves. It doesn't matter when or how that plot was revealed - just that it is the plot you have showing when the challenge resolves.

So to answer your question - when you reveal a new plot by a card effect, the only thing that happens is that your revealed plot changes. The stats and text of that plot are what you use going forward, but it doesn't change anything that has already been done or do anything "going backwards."