My dad had built a sieve using 1/2" hardware cloth that I still use. It is literally an antique; I'm almost 65 and he probably built it some 30 years before I was born! The only thing I use it for is my compost. I set it atop my wheelbarrow and shovel out the compost from the bottom. Anything that doesn't fit through goes back on top of the pile. What comes through the screening is absolutely lovely to use.
As far as your situation, I would say that you don't really need to go any deeper than you already have. Most annuals don't have very deep roots, and perennials will find their way around what's there. My advice would be to build up. Mulch up leaves in the fall and add. Add grass clippings. Keep augmenting the soil that's there with available plant material. Mulch with wood chips. If you think about it, when you harvest food from a garden, you're taking out a great deal of mass. That needs to be replaced. My front garden used to be much lower. By regularly adding plant matter, the level has risen a bit while the quality has improved dramatically. Before it was just sand; now it's deep, dark soil. The one exception to this is that you will want to remove any dropped fruit so as to not carry over diseases, unwanted fungus or insect larvae.
The latest research says to not till. There is a lot of life in good soil, and by tilling it, you're disrupting the micro ecosystem that helps plants thrive. Good old cardboard boxes without labels and tape will also help by keeping in moisture and acting as a weed barrier. (I've read that you want to soak it so that water will be able to penetrate it.) When you till, you're bringing up fresh weed seeds. rather than pulling up plants in the fall, some even suggest to leave the roots of plants in the soil to break down, in other words, you'd cut them off at soil level.
In permaculture, you'll hear the term, "Chop and drop" which basically means that you cut down plants at soil level and leave the material right there. When I harvest rhubarb, for example, I cut off the leaf part and lay it down where I think extra mulch would be helpful. It eventually will break down and feed the soil, but meanwhile it's acting like a mulch and preventing unwanted plants from growing there.
The best advice I can offer is to not have exposed soil. Some sort of groundcover plant and planting in layers will help with this. I use strawberries out front, but I also have sweet woodruff among many other plants. Wood chips are a wonder and can be gotten from local arborists who are eager to get rid of them (usually for free.) They are great for pathways or for putting around plants. Another thing to be aware of is to edge your garden. You can spade out and down about four inches or so, so that nothing will cross into your garden bed like crabgrass. Personally, I edge most of my beds with hosta because it's well-behaved, densely rooted so little can get by, and it's once and done. It also makes it easy to mow any lawn.
I wrote more than I thought I would. I hope you find it helpful. You can research any of the afore-mentioned ideas: no-till, chop and drop, permaculture groundcovers, etc. Good luck on your new adventure! I've included a picture of the front garden just as I started it. You'll see the hosta and the mulch. I'll try to add another picture of it more recently after.