Best tools for Southern California clay-like soil
I've been digging up the yard for a good amount of time now and have run into obstacle after obstacle. First, the soil is a little awkward to work with. It's a little more on the clay-side. Not to the point where there's many cracks from it drying up so much. But the ground definitely dries out and compacts pretty hard. Second, there's an invasive root system running under the yard from the neighboring Jacaranda trees outside the yard fence/wall (some of the roots are as thick as 8-9" or so and are doing some damage the wall as well).
I've been slowly growing my array of tools - just picked up a new long-handle fiberglass shovel as well as a bow-rake. I'm also borrowing a reciprocating saw from a buddy and picked up some pruning blades for it. And I have a serrated edge garden hoe on the way. But I think I'll need some heavier firepower, notably as in a mattock. Especially to try to pick at the pesky birds of paradise plants that don't seem to want to die.
I'm wondering if I should go for a standard pick mattock, or if I should go for a cutter mattock (since there are a number of roots running throughout the yard... and I think I've only hit the tip of the iceberg on that too). What would make the most sense? I do have a small hatchet but the edge on it isn't very sharp - I need to figure out how to sharpen it.
Otherwise, are there any other recommendations on tools? I was also thinking about getting a pitchfork eventually. Would some sort of cultivator/tiller be good to use as well? Ultimately, I just want to break up the soil enough to where I can work with planting in it. My intention is to remove all existing plants (or move some to other areas of the yard at least) and start a vegetable garden. It's not a super small space but it isn't large by any means.
Any tips/advice would be appreciated!