I don't think this is the advice you are looking for; but here goes. I think putting a stone facade on that grey wall would transform this yard. THe grey stucco is a dividing line between your patio and yard. Stone or something more natural would blend the areas better.
Back where it says single rose plant you need a small to medium size tree. i would eliminate the bed at the top of the stucco wall and make a larger sized bed of mixed shrubs and perennials between the corner of the chain fence and the corner where the "single rose" is located.

A line of shrubs or perennials around the edges of your yard is not as attractive as a well thought out grouping of trees, shrubs and perennials in the focal areas of the backyard. From the pic you have posted, the focal areas seem to be the back corners of the yard. If you could post more perspectives it would probably help identify focal areas a little better.

Could you tell us about the gate on the right side of the picture? Can you put a small tree on that corner of the yard?

Interesting ideas. What kind of tree would you suggest for the corner where the rose is? The gate at the other end of the yard is for a dog pen that the former owners had, so I can take it or leave it and can do whatever around there.

Here are some more angles:
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I would get rid of the dog run. It seems to serves no purpose except to block off a corner of your yard.

For the wall, I would do what drtygrl suggests and reface the wall. At least fix it up so it looks nice. I would also put a ledge for seating on the top of it. It seems to just be calling out to be sat on except there are those beds that it would seem you're sitting in.

I would also consider a path leading from the patio where the orange and green hose is. It's hard to really see from the photos, but that seems a natural location to get from the patio to the yard.

Yes my immediate project will be to fix up the wall. When you mention a path, where do you have in mind it going to?

Do you guys have suggestions for types of trees and shrubs to use in the corner as suggested by drtygrl?

If I do put a tree and some shrubs in that back left corner, would it be too much to put a fountain or large urn type thing in the middle of the square bed in the back and surround it with plants? I am also thinking of raising that whole square bed in the back and maybe rounding off the corners a bit more.

Also, if the dog run is removed, which I think is a good idea, what do you think I should do with that corner?

Thanks so much for the good ideas.

I think your previous owners were suffering badly from perimeteritis. Removing a lot of those beds, even in your mind to start with, might be a winning move. Start with this: where and when do you need shade in summer? Where would a larger tree have to be to provide it?

Is that a view beyond your backwoods there? Is it possible or desirable to see more of it? Can you make a place to sit at the back of the property that there would be some reason to go to? You want to design places for people, so imagine yourself at different spots - what would you do there, and how would you get there? Then, where would pleasant plants have to be to provide enjoyment while you are there?

Using the topography to make tiers or something along those lines across the yard could make so much more of this yard than simply using the perimeter.

I don't think the wall is ugly so much as it is an improbable colour, and it does look unfinished. A bench top or something like that: good idea. Stone facing would take up some patio space, how about slate tile?

Karin L

Just to further explain one thing, I would consider grading the yard according to the steps in the wall, making terraces that go perpendicular to it. Maybe just for a short distance, but enough to make the wall more integrated with the slope.

Karin L

The houses here are very close together, so the back of the yard just has a view of he neighboring home. I'd prefer to have it blocked from view. Except for the left side of the yard it is quite shady.

I agree the perimeteritis is an issue. I'm tryin to make my mind erase the current layout so I can start from scratch. At the same time I don't want to reduce my grassy area too much. I think the thy hand side of my yard could be changed a lot especially since it is so shady the bass doesn't grow well in that back right corner

How do you guys feel about this as a potential improvement for the back left corner. The picture shows euonymus wintergreen, Japanese maple, and chinese snowball. It also has some canna I am likely to plant along the back fence. Any suggested improvements?

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I this were mine and I had free work force available...
I always feel uneasy with a view that slopes sideways. So I'd move the table and chairs to the right - to the other corner of yard - and redo the retaining wall so it runs diagonally, at a 90-degree angle to the sight line from the highest point of yard to the new location of table. Maybe make that wall slightly curved, surrounding the new patio. I'm writing this without seeing your house or entrance to the yard so might be completely on the wrong track. Agree with suggestions about wall material, would also try smoothing the jagged upper edge of wall. Maple looks nice, but could use a bit more distance from fence. The cannas I would rather have near the patio, and in their place use some dwarf shrubs.