I know this kind of soil, too well.... It will take some work the first year, but then things will get much easier, I promise.

Are there plants you want to save? Dig them out and pot them up for now. Shrubs or trees that look healthy can stay. Just work around them.

For starters, get the black mesh out. If the old mulch on top is some sort of rotting bark or shred, rather than rock or color-died garbage, just dump it on the soil surface.

For your soil, get some topsoil and even more commercial compost, about one part topsoil to two parts compost. Then you dig it in and mix it up. (That's the hard work this year...). Watering the area first makes this easier. Go down at least a full shovel-depth, and mix the topsoil and the compost in well.

From the description of the current soil you gave, you probably want enough to cover the area with at least 3 inches of topsoil and at least 5 inches of compost.

Then, plant whatever you want.

If you try to just build on top of this soil, it would need to be very deep raised beds. But working the soil as above lets you then just do what you want to do.

I have clay soil and I grow veggies in raised beds and in soil that I've worked plenty of compost into, and the ones in the raised beds always do better. I also have lots of gophers and the raised beds are easier to protect from them, which contributes to their success I'm sure.

Wood beds need to be replaced after several years but concrete block or rock will last for a very long time. They also have the advantage of holding more heat from the sun and warming the soil, which can give your plants a head start in the spring. Although not all plants like to have 'warm feet.'

If I were you I'd do raised beds with nice, wide paths in between for access.

"I am really not very fond of mulch in the first place..."

We used a rototiller to incorporate the compost. We dug into the soil at least a foot deep to turn the beds under, after scraping off the grass (and weeds) on top. Then we added mulch (you can save the old mulch on the side until this step, it will break down and disappear eventually), at least 2 inches deep on top of the soil, around the plants.

The mulch is key to keeping down weeds, and keeping the soil temperature even. There are several different kinds of mulch to choose from, but you want the kind that breaks down (organic). The difference in my beds that aren't mulched is huge; I have to weed and water all the time!