What area of the country is, and what type of soil and drainage do you have?

Woops! Edited to include, we live in Arizona so it's very dry and hot!

Budget? Is there a pool in your future?

Do you have a bird's eye view plan of your space with measurements?

Yours is a big question that merely invites many more questions to get to the best use of your space. Trees planted further south of where you want some shade will be useful.

I was in Phoenix last weekend and enjoyed a much smaller backyard space with overhang from the house (it was 2-story with a deck above), a small pool, 3 or 4 mature mesquite trees of different species, and a patch of lawn that the owner said is "too expensive" to maintain, but looked fabulous. There were many birds visiting the trees and bird feeders, as well as large lizards hanging out on top the block wall. It was a small, simply arranged back yard space, nothing complicated, but probably not cheap either. However, over the span of more than 20 years, it has served the owners and their family well, and it still looks decent.

Are you one who enjoys working in the garden, or would you prefer to sit back and enjoy it? Are there children to consider? I have more questions than answers.

You should decide if the brick barbecue is an asset or a deficiency. Sometimes, they are missing parts and if properly fixed could be useful and enjoyable. Given that it's away from the house a bit, it would be useful only if you can take advantage of the campfire/firepit ambiance. A distinct disadvantage to its daytime use is the lack of protection from the sun. You have to figure out if this, with alterations, could be made useful to you. If not, demolish it and return the area to a clean slate. (If covered with a roof or pergola, directly or near, and with some paving underfoot, I could envision it with an old world, getaway, smokehouse feel for larger family gatherings. That wouldn't preclude you from also having a more convenient lounging & barbecue area directly at the back of the house for everyday and smaller gatherings.)

There is plenty of space to expand the existing patio if desired. Fortunately, Arizona doesn't have a law that a pergola must be flat. It can be designed to coordinate with your existing roof line. At any rate, the patio will not really be useful until it gets some shade. You'll need to decide if this comes from plants, an architectural structure or a combination of the two. (Are there vines that grow there?) Narrowing down a budget will help guide this decision.

I would consider some larger conifer type evergreens for the back, right corner of the yard in order to screen the utility poles and convergence of wires. Other trees, including palms, would be helpful in giving anyone using the back yard the feel that they are protected and sheltered.

The small tree at the back of the house seems neglected. You should begin limbing it up with the goal of eventually being able to walk under it as it gets larger.

I would really build a large covered patio off your home. If you don't want it to look odd, you could mimic your roof line with the patio. Then, I would take out all the grass and focus more on the natural landscape of Arizona. Something similar to this....

Hope this helps or inspires!

I don’t know Arizona very well but it seems to me that borrowing from Spain would be the best approach to the yard. So stop fighting the ridiculous grass and ditch it. Using glos advice go for hard scraping with patio pavers (or if on a budget make your own forms and pour your own concrete). And either pea gravel or similar for the balance. Yes use native planting’s (palms! Cactus!) it’s a sweet little bungalow and a generous yard so take your time and plan the space. I’d love to see some generous sized clay pots added to bring some Mediterranean/Spanish flavour too.

how does the front yard look?

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how long do you plan to live there?

start with a path.... lay out a loop in your yard, with just gravel, not fancy. then you can begin to upgrade it one section at a time. don't skip the lighting as in the cool of evening will be the best time to enjoy it.

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Wow! Thank you for your suggestions.

I think the backyard is about 2-3k sq ft, but I'm not positive on the dimensions. We are thinking of a small-ish budget (think $X,XXX or very low $XX,XXX). There aren't children to consider (at the moment) just two cats who like to escape to the backyard sometimes!

Our primary use would just be somewhere to relax in the shade and eat dinner/grill. We aren't really thinking of a pool at this time.

I'd love to add some turf and maybe some other landscaping just so that it's not a barren weedy/dry rectangle. My best bet may be finding inspiration photos.

As far as shade goes, I guess I just want input on whether something off the back of the house would be the best option, or because of the gable roof something built away from the house in the yard.

Something built off the back of the house will eat up your budget unless you are a carpenter/builder. A detached structure if simple enough could be a DIY venture with some handy friends to help, and if you are inclined or skilled enough. With the asymmetric shed roof of the house do try to mimic the more modern 60s vibe by going contemporary. No Victorian gazebos here... it’d look dumb.

...on the other hand a larger attached structure could add considerable value to your home depending on the neighborhood and sales.

Think more of this but on a smaller scale

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But a budget conscious pavilion could look great if done on a larger scale to fit the space

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Post again to show your project!

I would start with a patio and a free standing cover that you can eventually build a fireplace, pool, grass and play area around. Plant lots of trees and don't forget lighting for relaxing evenings out in the yard. Here's an inspirational photo from Royal Covers of Arizona: