Okay... this past year I went camping with two adults, a toddler and an infant, so I know what I'm talking about:
Firstly, the infant is the biggest problem if it's gonna be cold at all. When we went, it was in the low 50's at night. Not very cold, but cold enough. If you're trying to go on the cheap, there aren't any options for sleeping gear for an infant that I know of. There are sleep setups for infants that cost a bit more than we wanted to spend for one camping trip. We ended up putting our infant in a portable crib and she slept in a heavy winter one-piece outfit thing and we put her in a little sleeping bag. She was warm enough, but we were worried about her for the first couple nights.
The toddler sized sleeping bag we got from Walmart would only be good down to maybe 60 degrees. We got a fleece liner to compliment it, and that worked nicely. She slept on a foam pad.
Adults need what adults need. That part is easy. If the tent is big enough, get some cheap air mattresses for the adults.
As for the non-sleeping gear: I'm a fan of big tents. It's very convenient to be able to put all the bags inside the tent along with a lot of room to move around. We got a massive 10 man tent from Costco that was about $140. It's good... unless it rains. Luckily it never rained on us. I don't think they sell it any more. Stay away from Walmart brand Ozark Trail tents. Tried one recently and it really leaks like crazy. I think the Coleman brand should be good enough for what you want.
We also took the following:
7 gallon water container:
Coleman Twin LED Lantern (the single LED one is not bright enough):
Two burner camp stove:
A Pack-n-play for the infant. Be sure to get some mosquito netting to put over the top or the kid will be bug bait.
A portable toilet to put in the tent for the toddler.
LED headlamps for everyone.
Camp chairs for everyone, including a small one for the toddler. We used the infant's stroller instead of a high chair for feeding.
Everything you need for s'mores including extra long forks so the toddler doesn't get too close to the fire.
The rest of the stuff was just clothes and food.