I'm not an expert but I've learned a lot over the years through careful research and trial and error.
Calloways will test your soil for free. If you need to add soil to raised beds, Miracle Grow Organic gardening soil is great, or you can make your own mix with cow manure, peat moss, humus, top soil, and compost, all of which can be found at Home Depot and Lowes.
I would try the tomatoes in the shadier beds if they are getting at least a good amount of morning sun. Most veggies need full sun.
In my beds I already have planted onions from bunches, leeks, lettuce, spinach, and broccoli from seeds.
Probably next week I'll get around to planting tomatoes (look for thick stems around the base rather than tall blooming plants) and a variety of peppers which I'll purchase as small plants. The first week of April, I will plant (from seeds) okra (try Burpee's Baby Bubba Hybrid; it only gets 4 ft tall and is beautiful, you can find on the burpee's website), egg plant, cucumbers, green beans (choose a bush rather than a pole seed unless you have something for them to crawl up), squash, and a second round of onions. I only plant carrots in August to harvest during the winter because they need the cold to make them sweet.
My strawberries were planted last spring and have expanded via runners; they are blooming now.
Herbs do well here. Plant basil and cilantro soon from seeds and they will reseed themselves each year. Rosemary, mint, and lemon thyme will grow all year round. My parsley plants usually live about 2 years before flowering and dieing. Chives can be planted from seeds now and will reseed as well as usually live all year round, too.
You can mix in some marigolds or nasturtium (again look for a bush variety) to add color and repel insects.
Most years my squash don't survive thanks to squash vine borers but I'm not giving up! This is only my second attempt at broccoli, the first time I think I planted it too late and was unsuccessful. But after watching a friend grow big beautiful crowns, I'm trying again. And this is my first year to try egg plant and leeks but everything else I've listed has grown successfully and easily in my own garden and there is no reason it shouldn't grow in yours as well.
Be careful not to crowd your plants or you'll end up with low production, fungus and mold. Try to keep water off the leaves when you are watering, this causes mold and fungus as well. Cherry tomatoes and peppers will grow well in containers if you need extra room outside your beds. I try to keep it mostly organic but snails are my worst nightmare; they destroy the seedlings as they come up so I always use a little snail bait as soon as the seedlings emerge.
Good luck and happy growing!
