I'd second whoever mentioned hocking hills above. I've done a lot of camping, and would recomend that area for someone looking to get out a bit for the first time without roughing it too much.
You can somewhat have your cake and eat it too there, "car camping" by pitching your tent, but having your car nearby so you can bring coolers and such things that you wouldn't necessarily carry to more primitive spots. And, this being an especially important amenity for first timers, they have regular bathrooms with running water.
Of course, you'll be awfully close to other campers, especially during peak season. But it'd be a good way to get started.
It's also nice because there are a variety of trails to hike there, of varying lengths and difficulty, so you can really tailor a trip to fit your family's interests and abilities.
As for looking for equiptment, one place I highly recomend is Campmor (www.campmor.com) you can get some really good deals there, and they have a very wide selection of stuff.