I have camped twice in France alone with my DD 2.5yr old first time and 3.5 yr second time. Weather was mixed and had some heavy downpours second time round but we loved every second! I only have a small hatchback too and managed with clever packing (and DD squashed in the back in her carseat!) to take loads both times including tent, beds, duvet, toys, books, fridge, food for 2 weeks! gazebo, camping stove, bottle of gas clothes, elec hook up etc
I have a Tesco cheapy tent (£35 4 man) which i back up tarpaulin slung over if it looks like really heavy rain - okay we look homeless
but it works!
You can bag a cheap cooker from B&M, tesco or the likes (we went for top of the range-ish camping Gaz one and was about £50 and is fab - can do toast on the grill which was a must for me!)
I take a blow up bed (put some yoga/camping mats under it for insulation - and use my duvet. DD sleeps in with me and she is always too hot and thows all covers off and/or strips off!
Food wise we do a mixture of eating 'in' at the tent and eating out on campsite restaurant - if you don't want to buy a stove could just take BBQ - so long as campsite allows!
Camping is fab for getting away from it all and just reconnecting with your child/ren and I definitely recommend it. The first trip is the dearest cos of all the outlay on new stuff but once you have it each trip is cheaper obviously.
If you are unsure maybe book a site with lots to do - Haven have a fab deal on at the moment til end Jan think it's something like £6 a pitch per night and you can access all the entertainment, indoor pools and restaurant etc.