OK - so OP you've clarified that you'll start off car camping. Great, that simplifies the matter a lot since your gear can be the inexpensive Coleman / Cabella's / Walmart / Target / Big 5 / Sportco type (e.g. heavy / bulky). Note that when you take up backpacking, you'll have very little overlap in gear from the car camping stuff.
As far as where to find the stuff - as others have said, CL is a goldmine. Specific other places to look include Play It Again Sports (5 stores centered around Seattle, another in Vancouver, WA for SW / Portland) & 2nd Ascent in Ballard for used gear. I don't know what's in Eastern Washington as far as used gear shops.
Another Washington specific place to go it to
www.NWHIKERS.net and their gear forum - there will be a LOT of folks there that can give specific advice of where to score used gear. Lots of folks there in the Buy / Sell forum could probably hook you up with 2nd hand gear at used prices.
Now -for the specific questions of what you'll need for car camping and what to get that is inexpensive. Generally I'll recommend the Coleman type equipment for summer car camping as it's adequate for the conditions you can expect and, within reason, weight / bulk aren't an issue when car camping, unlike when backpacking, bike touring or kayak camping. I'll assume you'll cherry pick a bit on the weather and NOT go out into a week of solid heavy rain like we can sometimes get in August in the Cascades.
Shelter - get yourself a large, dome Coleman type tent that will comfortably fit your family and allow you to stand to change clothes. These are inexpensive and you won't care about the weight / bulk since you don't have to carry it. Cut out a sheet of plastic (painting section of hardware store) to be about 2" smaller in each direction than the floor and use this as the ground cloth and moisture / vapor barrier under the tent when you set up. Even in dry conditions, the ground has a lot of moisture that can wick up into the bottom of a tent. No need for fancy "footprints" when a cheap sheet of plastic does the job well.
Sleeping Bags - get your family the cheap, rectangular Coleman types. They'll be more than adequate for fair weather summer car camping. If you're interested in going in more marginal / colder conditions, consider "proper" (e.g. mummy shape, down or high end synthetic, higher end) sleeping bags. Once you take up backpacking, buy the best bag you can afford - a high end down lightweight backpacking sleeping bag is a "buy it for life" item. The one I bought back in 2001 (Marmot Arroyo, 30 degree, 800 fill down) has 3,000 miles plus on it - I used it on my PCT thru hike and it's STILL good to go for many more years. Store sleeping bags uncompressed else you'll destroy the down's ability to loft and loft = warmth.
Sleeping pads - here is where I'll suggest spending a little bit of money to get some comfort - well rested family = happy family. Go back to REI and check out the lower to lower mid price point inflatable air mattresses. You won't need the high end back packing ones. The GF and I got some comfy ones we use for car camping and kayak camping (they're heavier than the foam ones we use for backpacking, but take up less volume under the hatches, even though they're heavier). Be sure to store them with the valves open so they naturally inflate - under the bed is a good location. Only squeeze out the air when packing for a trip - open the valves and store under the bed when you get home. Or just get the cheaper Coleman type inflatable at Target / Walmart.
Cooking gear - 2 burner Coleman propane stove - inexpensive, bulletproof, easy to use. Use the 16 ounce throw away cylinders. Find it used or buy for under $40 new at Walmart (I just googled that). As far as pots / pans - that's what your kitchen is for when car camping. Break out your typical fry pans, pots, spatulas, forks, knives, mugs, etc. If you're a coffee person, one specific piece of gear I'd recommend for car camping is a French Press - look at Bed Bath and Beyond or similar places for a plastic one - they are a good substitute for the at home drip.
Lanterns - I'd suggest a couple of LED lanterns and headlamps for everyone. Skip the propane / white gas - batteries aren't that expensive and are far less of a hassle that futzing with mantles and fuel. Headlamps are easier to use than flashlights - hands can focus on cooking, etc instead of holding the light. Get cheap headlamps for the kids. Get nicer ones for the adults - lower end Petzl are a good choice (shop on the REI website, buy elsewhere), or the cheap ones at the check out at Lowes. Get one / two lanterns for general camp lighting. Note that if you're doing summer car camping in Washington, it's light out until 9 and gets light at 5 in the morning in the middle of summer, so you won't need too much light.
Other than the above, you shouldn't need much more purchased gear, especially if you're doing the State / USFS Campground route. Perhaps some folding chairs. All in, even going new at Walmart / Target for most of the stuff you shouldn't spend more than $500-$700 to set yourself up pretty well.
Large Coleman 4-6 person dome tents can be had for under $100 at Walmart. Bags, $25-50 / person. Stove, $40. Headlamps / lanterns - under 100 for the whole family. Saw coleman pads for $25 / each. And that's new.....check C/L and other places for used.
If you're doing dispersed forest camping, some of the blue collapsible water jugs are handy. You'll want a trowel / small shovel for digging your latrine / cat holes. Hand sanitizer, of course.
If you're going to take the kids on day hikes I'd suggest they all have size appropriate day packs - cheap book bag types are more than adequate for kids to carry a jacket, small water bottle (hint - one liter water / soda bottles work very well and are very light and tough - skip the purpose water bottles - heavy and expensive), snacks, their headlight. Get each of the kids a whistle (kept on their pack or on a lanyard they wear) and teach them to stay put and blow the whistle 3 times (loud) then count to 50 and repeat until they're found while they turn on their headlight to flashing mode (if getting dark). Don't load the kids down....just their jackets, water, snacks / lunch, light, whistle.
