@RobCam Car camping across the west and finding free “sites” is relatively easy to do. There is so much public land in the west that it makes it easy to find a place to park although sometimes you may have to drive a little out of your way to get off the main path. I’m typically at a new site each night so fast set ups/take downs are important.
I don’t know anything about camping with small children so I can’t speak to that aspect but I’m assuming you’re talking about 3-season camping/backpacking. For a general three season camping/backpacking tent (without getting into the topic of lightweight cottage manufacturers) I really like Big Agnes. I’ve had numerous tents from many manufacturers and for general three season use (with occasional snow loads) for car camping I would typically use my Big Agnes Copper Spur UL2. Quick to set up, reliable, and relatively lighter while still being able to stand up to wind and low snow loading. Literally hundreds of nights spent in mine over the years in the American west and Canadian Rockies. Big Agnes has a good product line that finds a nice balance between car camping and backpacking. Just do some research with any brand within the different models because there are differences between a $300 tent and a $400 tent even within brands.
Oh, there’s a really nice app for iPhones that I use sometimes that will show you where a lot of National Forest campsites are. It is called Allstays Camp & RV. Hope this helps some.