Researching where you'll be staying is the key to a positive experience. In general, you'll need to pack differently depending on whether you're driving in to your campsite or hiking in to the backcountry; among other considerations, backpackers have to limit the weight and volume of their gear. Get on the phone, go online, or look in a guidebook to find out about your campground-to-be.
Water and facilities. A developed campground in a national park may provide flush toilets, showers, and toilet paper. In the backcountry, it's dig-your-own-latrine time. Know before you have to go. Find out if the running water is potable, but bring drinking water and a filter or some iodine anyway. Add to your list things like water bottles, a water purification system, iodine tablets, and a trowel.
Weather and critters. Check the weather before you go, both what's usual for the time of year and what's predicted. Going to bear, tick, or poison oak country? Add to your list things like rain gear, calamine lotion, a bear bag (a bag to store your food in--and to keep it away from bears) and rope.
Maps and guidebooks. Make sure you have clear directions to your campsite. Study guidebooks, road maps, and topographical maps before you go, especially if you're hiking in. Bring them with you, along with a compass.
Fees and permits. If you have a campsite reservation, bring the confirmation with you. Want to hunt or fish? You'll need a license. Some ecologically fragile or heavily used campgrounds have rules about what you can bring to the area and what you can do there, so call to find out the facts before you show up. Backcountry hiking usually requires an access permit, which should also allow you to gather firewood and build campfires. Remember to bring enough money for fees.