Hello, friend! That’s an EXCELLENT question, and I’m more than happy to give you a few pointers! It’s excellent that you and your family want to start a small garden; gardens are something that I think everyone should have in their backyard.
There’s two things to keep in mind when you’re starting a garden: they take constant upkeep, and they take space. With those things in mind, here are some suggestions:
Unless you put down mulch or landscape cloth, you’re going to get a lot of weeds. Dedicate some time every week to yanking weeds you don’t want if you’re not going to mulch with anything. (Mulch is wonderful for little guys like earthworms and earwigs, though; they’re natural decomposers and mulch will also give spaces for beneficial beetles and spiders to hide out.)
You don’t need as many plants as you think you do. Two tomato plants, two squash plants (different varieties!), two cucumber plants, four or five lettuce plants, three bean plants, and four or five pea plants are excellent for a family of four. Some things, like celery and potatoes and carrots, can be grown at home but they’re a little trickier and can be hit or miss. (Potatoes and celery, for example, take all season long to grow, while carrots need sandy soil to grow well.) If you know what kind of soil you have you can tailor your garden to what works best!
If you know you don’t have time to dedicate to weeding and you’re looking for something fairly low maintenance that will attract pollinators and still be edible, take a look at edible flowers and herbs. Bumblebees and honeybees especially love Thai basil (and Thai basil is also much more disease resistant than Italian basil! Baker Creek Heirloom Seeds is one of my favorite places to get seeds for my personal garden, and at the farm I work for, we use Johnny’s Selected Seeds almost exclusively. Both have great selections of edible flowers and herbs, and both are more low maintenance than a lot of veggies.
If you don’t want to start from seeds, or don’t have the space to start from seed, always look to see what local hardware stores or garden centers have before hitting up the local Home Depot or Lowe’s. Big box stores like that take very poor care of the plants, and many of them come in with plant diseases; once you introduce these to your back yard they’re almost impossible to get out. Small little stores tend to take better care of their plants and actually give a shit about the people buying them.
Another big point: If you grow veggies, you’re going to get gross little predatory creatures like aphids (very common on lettuce and other greens,) cabbage worms (fat green caterpillars that like cabbages, broccoli, kale, and other brassicas,) and slugs (wherever it’s wet.) You can control these with beneficial insects like ladybugs, who will devour your aphids, or you can spray a little soapy water on them (Dawn is the only liquid dish soap approved for use on wildlife, and it’s also safe for use on plants as long as you’re using the original blue stuff!)
I think those are the best pointers that I can give you as a first time gardener; feel free to DM me with any other questions you have! Most importantly: have fun with it! It’s a great hobby that has such a profound impact on both you and the world around you!