Tomatoes and peppers need a longer growing season than I can provide up here when it comes to growing by direct seeding so those are always bought. I've tried starting bell peppers early under grow lights, but it just never is the same. They get too leggy. I probably need better lighting.
I never use chemicals, and I've had issues with powdery mildew when it comes to cucumbers and squash/pumpkins. My best success with cucumbers is letting them climb a small trellis. Having the vines in the air promotes circulation and keeps them dry. I tend to run very wet here so this may not pertain to you.
Kale grows here like a weed and I do an entire bed of different varieties now. Everything in that bed goes to the chickens. Spinach grows well, but many varieties tend to bolt. I can never seem to get brocc established before something goes awry with it. Most of your mixed salad lettuce varieties grow well here.
Most root veggies end up stunted for me. Either my soil isn't loose enough or too wet. I've tried carrots, sweet potatoes, and beets and they've all ended up poor. Sweet potatoes definitely prefer a warm climo. I may try them again, but do them in a raised fabric pot so they can be warmer and drier. Then at the end up the year I can just dump them out to harvest them.
I've gotten more into fruit trees/vines lately and most of the garden beds are literally for the birds.