A stone patio will look great! Another idea is to do a seating area under your deck. Check out this example.

Our yard sloped to the point that it was almost unuseable - we ended up putting in a retaining wall and filing in from the house to the wall to make a flat surface both above the wall and below it...the top is grass and the bottom was play area suitable. Since the kids are grown now, that bottom area has some shade and thus has herbs and wood for fires, etc. Might that be an option for your yard?

Wisteria or trumpet vine on the columns. It looks like you need shade, so get a shade tree to provide some shade for your patio. You don't want anything that will end up too big. I'd go for something decorative, like a dogwood, magnolia, or flowering cherry. You can also try fruit with a grape vine up your columns (you want to check the sun requirements) and apple, pear, cherry and plum trees are very pretty, as are quince and medlar, but their fruits are less popular.

My husband definitely wants to use the underside of the deck as a sunroom and screen it in. Thanks Granite Grannies - what a lovely space in your photo. The yard faces south and the idea of a tree is a great one we also plan to do here. We love Japanese Maples. The slope does make it unuseable now so it will need to be graded to put in a patio.