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 the tent over tarp idea.  Well, out here a tarp is doable because at night bugs are rarely a problem.  But on the east coast anytime in the warmer months the bugs will be an issue.  A tent, or even just a bug shelter, is essential.  Oh, and just to let everyone know, the REI Trekker sleep pad 
(1.75 inch loft) is on sale for $39.  Thats a great price for a good solid pad that comes with a repair kit.