What is the best campsite within a few hours of brisbane?

May be a bit further than you want to go but Cania Gorge is good for kids.
Excellent camp kitchen, lots of wildlife in and around the camp ground.

Try Neurum Creek Bush retreat, up near Kilcoy, has everything you've asked for. Kids can run feral, take their bikes.
Take your own firewood as his is a bit exxy for price and didnt burn that well last time I was there last year.

http://www.ncbr.com.au/index.php

Cheers

Pete

Try Big Riggen at Rathdowney - great for first time campers, flat, green grass - all the rest and on the banks of the Logan River. There is also Flanagans not far, just around the corner.

Big rigging (2hrs from north side of Brisbane) is a great spot, Midginbin Hill is a bit further away just over the border (3hrs), Kenilworth is a great national park so is Jimna Very Cheap! and free fire wood $2 showers, and last but not least there is a place up near Imbil fantastic spot for the family great showers really great swimming spot hire canoes and so on and on long weekends they have Karaoke for the kids which is quite fun to watch, deer feeding every afternoon and a great playground but book early it gets quite crowded up there (this would have to be favorite spot) and dog Friendly!!! which is a bonus !!

I would say without a doubt amity camp ground on straddie would be the best with 4 young kids . Powered sites are only for vans etc but the large undercover bbq area has power points . Stainless benches are about 3 mts long with ample tables . Shower block and toilets are great . Small kids playground , Netted swimming enclosure , Jetty for fishing for dad . A yearly beach pass to drive on the beach is $35 . The straddie barge is about $120 with racq card . http://www.stradbrokeholidays.com.au/index.php

if you don't mind the bit extra distance then another vote for Cania Gorge. stay at the Big 4 van park which has it's own bird feeding each arvo just like Currumbin Bird Sanctuary. my then 4 year old daughter raved about the place for ages after we went there.
it's bitumin all the way there, has the gorge to go walking in aswell as the dam to try a spot of fishing. if you have a 4wd then a whole day is taken to visit the crashed Liberty bomber plane wreck in Kroombit Tops Nat. Park.
Park has 2 camp kitchens and even used to do a free morning tea twice a week (damper, cakes and tea and coffee). lots of powered sites aswell as cabins and plenty of grassey area for the kids to run wild.
we were told about this place by friends who visited there and have since reccomended to many others. definately a great spot for people with young families.