I also would like to know if the space behind the double doors is kitchen or living space. More pictures of the patio would be helpful. Pictures of what is further in front of the doors and to the right as I am looking at the picture would be helpful. From this picture, the nook to the left as I am looking at the picture might be a good space for an outdoor kitchen/bar. Define a dining area, a lounge area, and maybe a fire pit if there is enough room.

The space inside is living space/office space.

Some lounge chairs achored with a colorful indoor/outdoor rug. String some lights from under the rafters.

If rain is a problem dripping through the deck above consider adding some roofing that would protect your outdoor furniture.

A firepit would be nice for evening gatherings.

Warm up the space with pots of shade plants or whatever is appropriate with your sun exposure.

Don't forget plants in colorful pots

A palm fills in spaces.

I am struggling with the same problem. Thought about a fire pit as leelee suggested here, but concerned that it might be a fire hazard if placed under the deck-- does anyone know?

How about a hammock hung between the supports?

There are deck systems that go under the decking to provide a dry area under the deck. One is called "Dry Below". Once you have created a dry space, you can add furniture to create an outdoor "living room". The cheapest way to cover the concrete is with a stain. There is also what is called "stamped overlays" for a stamped concrete look. It is a thin polymer that can be colored and stamped to look like brick, stone or tile. Or, you can have a spray texture applied to look like stucco or tile. If you want a more upscale look, try using a recycled granite (rough side up). This is applied with an epoxy so it set thin. And of course, don't forget the plants. Containers come in all sizes and you can even add trees and shrubs as well as annual flowers.

I agree with bringing lots of color out on the patio. It looks pretty drab right now. Anchoring each defined area with a rug and pops of color and potted plants will bring the patio to life! I would put the fire pit out from under the deck or put in a eco friendly fireplace as an anchor at one of the ends or sides.

These are the only other photos I have of it currently

Looks like you have room to put your fire pit out from under your deck and away from the end of your stairs to the upper deck. I still think the nook would make a good bar/kitchen area since it comes out from a living area. Make it more lounge feeling if you have dining on the upper deck. A great relaxation area just to hang out it!

There is so much space and potential between your upper deck and lower patio. Great suggestions above for that outdoor living space below....an extension of your living space behind the doors and a break from the office. You may want to include a play area for the children down the road, as your profile picture suggests one coming. Congrats! I am assuming upper deck would be for outdoor dining and the grill.

You could string some outdoor curtains up along the edge of the deck to give the space more privacy and make it feel like a room. It looks wide enough that you could hook up an outdoor tv and put some couches out there as well.