Wow that's a lot of yard. Definitely get some trees planted asap for shade.

Here's a primitive and basic stab at arranging the yard. You should draw a plan to scale and work it out in plan view. (If it's not to scale, it will not be very useful.)

I'd do a raised deck by the doggie door, or something similar so there isn't such a drop outside the doggie door. Even if they don't mind it now, they will when they get creaky and old. What you have now, but a bit larger and perhaps higher at minimum.

Do you think they will need a run, once things are planted? If not, consider a temporary enclosure (like a well-anchored exercise pen) until you get rid of the mud. Not having a dedicated dog run will make it easier to plan. You might, however, depending on the dogs, want to put a charming fence around the veggie garden.

Zelkova is a nice easy shade tree -- full size, vase shaped, small leaves. It grows pretty fast. It's only drawback is that it is not native (and I'm all about native plants for ecosystem sustainment reasons) but even if Zelkova's leaves don't feed our native birds and bees, it's branches provide shelter.

For privacy from side-yard neighbors, look at Southern Arrowwood. We have some in our backyard and they grow quickly into a tall thicket, butterflies flock to the flowers, and birds LOVE the berries.

For easy garden/decorative plants, look especially at the perennial sunflowers here: izelplants. I like that website because you can set various parameters to focus in on what plants meet your requirements.

For dog fun, consider a "pondless water fountain" like:

(sorry I don't remember where I copied this pic from)

You and your dogs are lucky, enjoy your new house!

(temporary side track: Gardengal, you're right, an individual's choice doesn't have to be limited to "native=good, imported=bad". I make room in my yard for peonies, daffodils, and I adore my metasequoia. But I choose new plants by this quote: "A plant is not part of the ecosystem until something has eaten it.")

Great advice to observe your dogs before finalizing plans :-)

Consider creating terraces.

Decide how much room immediately behind your home you eventually want for a backyard patio. Leave at least six feet beside each fence not dug but dig a ditch/footing for a wall of concrete block down to hard ground across the center of the rest of the back yard.

Once your terrace wall is built, level the area above the wall, including by adding a truck load of gravel and/or dirt and/or shoveling dirt from below the wall to above it to level both your patio area and a space below the wall.

Once you've reserved your patio space -- while your dogs pack it from using it -- you can begin to plant your garden, preferably in a way the dogs can be taught to leave what you plant alone.

Pay attention to how the sun passes over your back yard so you're not planting anything that will eventually shade your garden too much for it to grow.

Brackets on fence posts can hold potted plants -- choose the strong ones.

https://www.google.com/search?q=very+strong+bracket+for++holding+flower+basket&source=lnms&tbm=isch&sa=X&ved=2ahUKEwjy1KW5l4LoAhWiSt8KHRZkBw0Q_AUoAnoECAwQBA&biw=1366&bih=628

Consider the stronger tomato cages for tomatoes and beans. You can buy individual trellis/cages in the spring from the big box stores or build your own -- there are You Tube videos on how to build different kinds. This is just one.

https://www.youtube.com/watch?v=ozQNsd5Oek4 

Consider building a mini greenhouse using plumbing pipes and/or wood.

https://www.youtube.com/watch?v=KsSI4w3CqEo

https://www.youtube.com/watch?v=sQWPnaiixe0

https://www.bing.com/search?q=building+greenhouse+with+pvc&form=EDGSPH&mkt=en-us&httpsmsn=1&msnews=1&plvar=0&refig=d03dc932a900445e84174ddd49ea6bb8&sp=1&ghc=1&qs=AS&pq=building+greenhouse+with+&sk=PRES1&sc=8-25&cvid=d03dc932a900445e84174ddd49ea6bb8&cc=US&setlang=en-US

If you build raised beds with wood, remember to line the wood box with plastic to make the wood last longer.

https://www.youtube.com/watch?v=ef_TS19TRQg

and / or building tiered structures for plants

https://www.google.com/search?q=wood+tiered+structures+for+planting+strawberries&tbm=isch&ved=2ahUKEwiv8fyvk4LoAhXEEFMKHS1JBKMQ2-cCegQIABAA&oq=wood+tiered+structures+for+planting+strawberries&gs_l=img.3...95448.101579..102697...0.0..0.134.1724.15j4......0....1..gws-wiz-img.NbM-ukJ-fRs&ei=qFFgXq_9FcShzAKtkpGYCg&bih=654&biw=1366&client=firefox-b-1-d#imgrc=irj20Wqm9lLhnM

The RHS has done research on the wildlife value of natives and non-natives in gardens. The overall message is plant a variety and plant densely. Articles in these links.

https://www.rhs.org.uk/advice/profile?PID=970

https://www.rhs.org.uk/advice/profile?PID=1019

https://www.rhs.org.uk/advice/garden-health/wildlife/Native-and-non-native-plants-for-ground-active-inv

The yard is flat as a pancake. Terraces? Totally unnecessary.

Good articles, floral_uk, thanks for sharing. It clarifies (for me) the difference in attracting pollinators, "shrub level" bugs and "ground level" bugs (my terms, the article is better). I like the simple rules each article starts with -- sensible regardless of where you live.

I was going to say the same as someone above, I tried to put in a garden for somebody with 2 dogs, and I learned the hard way that dogs like to run the fence line, and these particular dogs liked to dig doggy sleeping pits. One of them peed A lot and marked everything, only the most pee tolerant plants had a chance of surviving. Another garden I was maintaining, the dog turned everything into chew toys, a once beautiful Japanese inspired garden just turned into a filthy shredded disaster. I guess it depends on the dog, but I would recommend going with very robust plants that will bounce back from trampling, peeing and chewing. Some of the plants that seemed to do better were not surprisingly, I guess, some of the more common garden plants you see in parking lots and public parks, weigela, burning bush, spirea, sea oats, Siberian or Japanese iris, rugosa rose, salvia, dogwood.

I'm not much help about the design, but before you put in the gardens, you might want to think about fencing that area off into a "dog run" of sorts while you get the grassy areas established and healthy (with robust grass). Our lawn guy told us its pretty much pointless to try and grow new grass in a dog yard, they'll just kill it before it's established enough to survive them, especially with two large dogs.