Coleman
Kelty
REI
Mountain Hardware
MountainGear
Mountainsmith
The North Face
Outdoor Research
Sierra Designs
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My $.02. Start with a tent, not a tarp. Why? Better shelter, less bugs at night, no waking up with a snake keeping warm with you... Tent, my friend (or a Teepee, since those are being pushed a little). If the kids want to do a lean-to or something, let them have fun. I'm not a fan of the open air myself. DO invest in a good sleeping pad!

I second most of this list. BTW, it's "Mountain Hardwear" not "hardware." It's a pun. I've got clothes by them, TNF (The North Face), some Outdoor Research, had a few Eureka tents (not bad for beginning). Don't discount the REI store brand, as it's pretty decent quality at a good price. Same goes for EMS (Eastern Mountain Sports, a REI-like, East-coast chain). Personally I despise Coleman gear. Other than their lanterns (which rock!), I've found their stuff to be marginal, at best. Also try Arcteryx for jackets and backpacks. MSR for camping stoves, Mag-lights (the mini-mag) for flashlights (they're not the highest end, but extremely robust. Mine is 15 years old and going strong).

Some of the links on this thread are real good; check them out! You can also find amazing discounts at sales; ask the sales reps when they have their scheduled sales, and when new equipment gets released. Like, right now you might get good deals on winter stuff, as summer is coming. The folks at REI are real knowledgable, so pick their brains. Have fun, stay safe!