For a first timer I would suggest that you start with radishes. they are normally the first harvest of the year and spring radish are normally very sweet, large and even though you will plant red radishes they will come in a variety of easter egg type colors in the early spring. As the season goes on they turn red and bitter like the ones you are use to buying at the store. You will have a new appreciation for radishes.
Next lettuce is great. Butter Crunch is a nice small lettuce that forms a loose head. its a bit like miniature romaine but better.. Its great in salads .. late spring early summer harvest… form a bed and then sprinkle broadcast seeds across the top on the front third of the bed.. as they start to grow and get about 3 inches tall use a kitchen spoon to scoop them out and plant in the rest of the bed about 8 inches apart.
Zucchini is very easy but takes a lot of watering. Kids will be impressed by the big yellow flowers and zucchini that seem to go from 5 inches to a over a foot overnight because of the watering. they are also good sliced and cooked in spaghetti sauce with Parmesan on top so they will like eating them. you will start getting them about mid summer. Make sure you space your plants well… at least 2.5 feet between plants which for me means one at each corner of my beds and one in the middle.
You will need a fence around your garden with no bigger than 1 inch holes.Local Home Stores will sell green fence that is coated and it will last about 5 years .. install it with the green fence stakes .. you will need a sledge hammer or large hammer to pound them into the ground. you can attach chicken wire to it to make the holes smaller if you can’t get tight green fence.. get a spool of utility wire to twist tie it to the green fence at the bottom or you will have rabbits in your garden.
Beans are neat and taste good and don’t take much work to plant but are difficult to pick. same with peas. Another easier alternative if you like them is snow peas they grow in a small bush and you get lots of them from a small area.
All dead garden material must be removed from the garden .. a mulch pile should be at least 10 feet away so it does not attract pests to the garden.. farther is better.
Final Note
Gardening with your children is more than just harvesting for the family. There should be some fun things that you incorporate such as Strawberries or maybe you can plant some grape or other fruit bearing plants.
When your kids grow their own vegetables they will be more likely to eat well and they will understand that a lot of work goes into having good meals on your table. Gardens aren’t instant gratification and sometimes you have failures. Its about learning and then applying that knowledge in the future.