We had a large porch off the back of our house and put a very long, deep commercial grade awning across. The awning had drop down sides, steel poles and attached directly to the house on a rail system. It was very cool (see image). We loved the feel of it being vacation-like, unexpected and extremely useful. You are low enough to the ground to do a patio in stone or brick. To save money I would consider the awning, though you do have to remove the cover in the winter if you are in a colder climate (snow). To maximize your back yard I would run the cover from one set of steps to the other and create an entire, functional outdoor room. Even in thunderstorms you can drop the sides and have a great space to hang out with friends. Good luck!

It looks like you already have a patio towards the rear of the pic (?) You could install a deck to be even with the front door and rear slider with steps from the slider side down to the back patio. A seasonal umbrella would provide protection from the sun, or if you want something more permanent a small scale arbor could work. See first pic below for an example.

Another option, is to connect the two stair cases with a brick planter box between them to echo the brick you already have on either side of the siding. You could then add a stone or brick patio from the the front door area clear to the back.

Cute house - I love the brick!

Below are some photos that might give you some ideas.

On more visual of a house that looks to be from the same era as yours. I like the simplicity of it.

If you do a patio, consider bluestone cement pavers - they look as good as slate but you won't have the maintenance issues.

I'm in line with the comments here. You could do a deck straight out from the sliding doors to the doors in the foreground, then have a solid concrete or stone patio in the lower level. A trellis type structure would give you some shades without closing in too much.