When you can! And the weather is nice enough. I'd start with 2 nights for a first trip. One night is also doable, but since you usually have to leave the site around 11am, you its a LOT of set up for not much time. Make sure you arrive with enough daylight hours to put up the tent and get oriented. And also to get a fire going and get people fed. We've arrived at night before and it kinda sucked, though we'd stopped for fast food so at least no hangry-ness to add to the mix.
The fun part! Decide how far you want to drive, and then find parks and look up amenities and reviews. We went to a place within an hour away for our first couple of times, since I was nervous, and knowing we could just come home if things went sideways helped me feel better. We look for water nearby in the summer, hiking/biking trails, etc... Think about what you want to do and make sure they have it. Some sites have fishing, boat rentals, pools. And I ALWAYS read reviews about cleanliness, maintenance of trails, noise levels. If you want to bring a dog, make sure there are pet-friendly sites (there are less, and tend to fill up faster---if you bring a dog, you can't leave her alone, so someone may need to hang back from swimming/biking so plan that carefully. I did not enjoy missing out on the pool on a super hot day last summer to babysit the dog). Look closely at the map of sites and pick where you want to be (not near the dumpsters, for example, or closer to the water). Site sizes vary greatly---even within the same campground---so if you need more space because you are going with friends or have a large family, pick a bigger site.
You need a tent and places to sleep. We started off with a hand-me-down 4 person tent, and have since bought a 6 person tent to have a little more room. 4 person=exactly that and we couldn't fit our dog in their with the 4 of us. We sleep on sleeping bags over pads (you NEED pads, or you will feel every rock!) but you can have air mattresses if you are fancy! Bring pillows of course.
We also have a rain cover for the tent. We have since gotten fancy and bought a tarp and shelter (netted on the sides, solid plastic on top, for rain/mosquitos).
Other big things we have: propane stove and folding table for the stove/cooking, chairs, a big cooler for food.
This is the hard part because you need to bring EVERYTHING you will need with you. Plan out your meals/snacks and bring all food and implements needed to cook and eat/drink, including water. So dishes utensils, soap/sponge, something to carry dirty dishes to the sink with, etc... Implements for starting the fire (firestarter, wood, kindling, lighter), trash bags, toiletries, games/entertainment, flashlights or headlamps, first aid, towels, clothes. We have 3 "camp boxes" and keep things that are solely for camping so we don't need to hunt for them each time. Make a list. Check it three times.
keep it as simple as you can to start with. for our last trip we did: burgers, hot dogs, quesadillas (on the stove), toast, cereal, bacon & eggs. We've made nachos in a camping dutch oven over the fire before and want to try dutch oven pizza. If you don't have a stove, you need to obviously stick with things you can make on a grill over the fire (and be confident in your fire building skills). We bring fruit. LOTS of snack food---you never know when the fire might take hours to start! S'mores ingredients (duh!) and lots of water, as well as juice, and booze.
I...think that's it. Questions welcome!