First decide what sort of campers you are likely to be:
'Wild' campers, favouring sites where you can have a fire, camp where you like in a rural / wooded site and have the kids entertain themselves by racing through the woods etc. Mostly these sites do not have electric hook-up (EHU), they often offer bushcraft courses for kids, and people tend to sit outside their tents round the fire at night. May well have compost toilets etc - but you will be a good distance from other campers. Option to cook on your campfire.
or
Traditional campsite campers: pitches in rows, lots of on-site facilities, maybe even a pool kids playground, almost never allow fires, often have a choice of a pitch with EHU.
This will affect the equipment you need, cooking choices etc. Electric cool box v passive box etc.
Then think about sleeping: air mattress v SIM is a very personal choice, SIM is much warmer, and doesn't need all that inflating, the rest is down to which you find more comfortable.
If you buy sleeping bags buy good ones: Vango or Coleman 2 season, for example - don't buy crappy Halfords cheap nylon ones. We almost never need anything other than our Vango sleeping bags and cotton pyjamas in the UK! this sort of thing
Chairs and a fold-up table.
For a tent, I would look at something likethis tent. Very roomy, easy to put up, good brand, will not leak, fully sewn in groundsheet. It will fit the 5 of you for now and then in a year or two your older 2 will want to have their own small 2-person tent, which is fun.
Something to cook on: the 'briefcase' stoves are a good cheap one-burner option to get started with- take 2 if you intend to do much actual cooking, and a portable BBQ. (personally I hate disposables - the coals are too close to the food, everything gets sooty or burnt, they burn down too fast, bad value).
Many of the 'wild camping' sites will hire you a campfire grill to use over the fire.
Check out the Camping Board - lots of kit and camping tips discussed in great detail!