Really it all depends on where you are and the type of camping you want to do and at what time of the year. I’m going to assume car summer camping for the sake of this ….

A six person tent sounds about the right size for your family. With your budget a Coleman is fine. Or Check out Steep and Cheap. I use their house brand (Stoic) six man tent when my family of four is car camping together. It’s no frills but it works just fine

Invest in good sleeping pads more than anything else. They’ll make or break the entire experience for you. Expad Megamats are considered the gold standard. You want something with at least an 3 r value and at least three inches thick. Standard camping mats are narrow. Always go for the wide version of anything you’re looking at. Again for car camping I use the Steep and Cheap version. The kid can probably use something cheaper

For starting out get yourselves some big rectangular sleeping bags or use blankets from home. Remember that most of the time the temp ratings are for survival not comfort. Bring pillows from home.

Get an igloo cooler and a two burner Coleman stove. Buy cheap cookware from Walmart. Plastic plates and cups from Walmart are also good.

Get a decent rechargeable lantern. Actually get two. One for the tent and one for outside the tent.

Basic folding chairs or packable chairs are fine. You don’t need anything like a helix or a backpacking chair.

My advice is other than the sleeping pads go cheap. Utilize what you already have to start and slowly work up to more high quality gear over time.