I have learned that Americans are so totally resistant to improving their gardening techniques that they produce a fraction of what they could produce in the same time and area. They insist on doing things just like their great-great-grandfather did, when he had to carry a musket to the garden in case he was attacked by Indians.
I study agricultural research, and I apply the improved techniques. In a community garden, my plants are at least three times the size of other people's, and they're wider, stronger, more robust, and more heavily loaded with fruit. I get between 5 and 10 times as much produce from each plant as the average person in the community garden. I'm telling you this so you'll understand that what I'm suggesting really, really works.
You're definitely right to stop old-fashioned tilling. Turning the soil destroys the mycorrhizal fungus and bacteria network under the soil. The network feeds and waters the plants in a symbiotic relationship. Every year you avoid tilling, the network gets stronger, and your production increases, if all else is equal.
However, one of the worst things you can do to a garden is to put fresh wood chips in it. When wood starts to decay, it steals every tiny bit of nitrogen that it can possibly get, making your garden as mediocre and poor producing as possible. It's too late this year to improve production by removing the fresh wood, but next year, rake it off and compost it. Once it starts to rot, it will stop stealing nitrogen from the plants.
I know someone, who has no gardening knowledge, so he tried using fresh wood chips to try to keep down weeds. The weeds grew better than his seedlings in the nitrogen-deprived soil, and he got very, very little production. I don't think he even got one single tomato.
I'm assuming that some of your plants are sort-of doing okay because you planted them as seedlings, rather than as seeds.
Any plants you plant as seedlings should be mulched with black plastic to totally eliminate weeding and to dramatically maximize production. Black plastic, as a mulch in a vegetable garden, is so wonderful that the prevention of weeds is a minor benefit. It improves your garden in so many ways. AND YOU GET A LOT MORE PRODUCE, with less work.
* It warms the soil in the spring, giving the plants a serious boost, which increases your production pretty dramatically throughout the summer.
* It saves water (which saves you work).
* It evens the availability of water, preventing periods of wetter versus drier soil. This is really important for your cukes, melons, tomatoes, and peppers. If watering is uneven, calcium is not continuously distributed to the nightshade and cucurbits, and the bottom of the fruit starts to rot.
* It reduces certain destructive insects. Some insects lay their eggs on your plants, then the larva hatches and goes into the soil to grow and produce new generations of bugs to attack your plants. The black plastic barrier minimizes that, and you have fewer bugs without using insecticide.
Stake down the black plastic with landscape staples. Plant first, then cover with black plastic, cutting an X for each seedling. Gently pull the seedling through the plastic. Place two staples by each seedling to keep them from slipping under the plastic.
That's it! The garden takes care of itself for the rest of the year, giving you much more produce than you would get without the plastic.
DON'T USE FAKE LANDSCAPE CLOTH instead of black plastic. The fake stuff, that they sell to amateur gardeners, isn't uv-treated. It also allows some weeds to grow. Get big sheets of black plastic at Lowe's. Drill some holes in it through multiple layers when it's still folded.
If you want to really improve production even more, you can try to buy a product called IRT. It's a very dark green/brown plastic that is even better than black. It still keeps down weeds, but it let's in more UV light in the spring. It is a university-researched product that professional growers use, so it comes in enormous rolls. But a few online retailers will buy the big roll, and sell you however much you want. (I think Pinetree and Johnny's.)
BEETS! Beets are one of the easiest things there is to grow. It's pretty impossible not to be able to grow them. It takes no fancy growing techniques, and you certainly don't need to start seeds early for transplants. (Transplants aren't very successful with root vegetables, anyway.) But you do have to weed them a little right after you plant them.
The best way to plant beets is by "square foot gardening" which really minimizes weeding. Old fashioned planting in rows doesn't work nearly as well. Scrape the nitrogen-robbing mulch off a 2-foot-wide section. Scatter the seeds about a half-inch to an inch apart covering this whole section, so you'll have a 2-foot wide section of continuous beets. Cover with a little soil. Keep watered. As they grow, a few weed seeds will also germinate. Initially, you have to pick baby weeds out by hand, which is a lot less work than fooling with transplants, they're not going to work well anyway because it's a root vegetable. As the beets grow, they will shade the ground, and you won't have to keep on weeding. You plant the seed so close together so the beets shade the soil. As you thin the beets, you have some tender little plants to add to salad or to cook with mixed vegetables.
If you insist on old-fashioned row planting of beets, mulch with black plastic right up to the row on both sides, and you will minimize weeds.