If you want to get your wife onside, keep her warm. Cold is the single worst part of camping and is to be avoided in the novice reluctant camper at all costs.
If money is an issue I would take airbeds and duvets, sheets and pillows from home. Put a warm blanket between your sheet and the airbed to act as insulation from the ground. Take more blankets for over the top (just in case) and full thermals for everyone. Plus lots of layers - the evenings and nights get cold when you are outside even in the height of summer often. Also a hat for everyone - it will help to stop a lot of the heat escaping.
If you can stretch to camping chairs for everyone that will make things a bit more pleasant than just sitting on picnic blankets. If you can't then at least one each for you and DW. We took an inflatable sofa this time (www.sportsdirect.com/gelert-inflatable-sofa-789135?colcode=78913599&src=google&gclid=CP3didvxh8YCFQXLtAodPQIAfw&gclsrc=aw.ds) and it went down a treat with all the children (and visiting adults too). Took a long time to deflate admittedly.
You will need a gas cooker. A single burner suitcase one (www.sportsdirect.com/gelert-portable-stove--787217?colcode=78721790) is fine if you aren't planning on doing much cooking or are going to mainly BBQ say. You can make a cup of tea in the morning at least - essential.
Maybe a picnic table if you have one - I wouldn't start investing in too much until you know your wife is on board. Ebay is filled with nearly new camping gear from people who thought they would like camping, and were wrong.
Some sort of tent lamp and headtorches are also a good idea - they are a bit uncool but leave your hands free to do other things which can be useful with small children particularly.
I would bring a bucket for night time wees, but you will know whether your DW is a pee in a bucket type person or not.
Definitely a potty for the children - you do not want to be traipsing across a field all night long.
A coolbox/bag to keep the wine milk cold.
Something easy to throw at the children in the morning when they say they are hungry and you still don't want to get up. We bring chocolate chip brioches for this purpose but chocolate croissants or similar work well too. Fill up water bottles for them the night before too.
Wellies for all and crocs if you have them - they are easy on and off shoes to stop you and them trailing mud in and out of the tent (we have a total no shoes rule in our tent).
Will you be able to have a campfire or a BBQ at least? If so, don't forget the marshmallows and kebab sticks for toasting.
Enjoy!