How do I control what payment method is shown to a user on a site? How do I make one show first?

When you are at the window at Sites > [site name] > Settings > Payments, you will see there is "Sort Order" field on each payment method enabled for the site. Go ahead and set a numeric integer value (1, 2, 3, etc) for those payment methods, that will control the order that they are displayed in on checkout. If you would like to limit certain payment methods to only certain users in the store you can select the "Allowed Groups" that you want to enable each method for and put your users into Site Groups.