Before you get to the camp site and start camping for real, it's better to try putting up your tent if you haven't already tried it out. This will ensure that you have everything you need for erecting the tent. It will also avoid the confusion of getting to the campsite, looking at the instructions and wondering what you need to do. You will also avoid becoming the entertainment for the whole camp site as you argue with your partner while struggling with tent, instructions, poles, pegs, guy ropes and other seemingly pointless parts. That was my first camping experience, anyway. If you have a choice, wait for dry and still weather before the first attempt at the tent. Wind will make everything ten times more difficult, and rain means having to dry the tent completely before packing it away, otherwise mould will start to form on it.
photo credit: abbybatchelder
Your first camping trip should preferably be quite close to home, so if anything goes wrong you don't have far to travel. It also means that if you've forgotten anything you're probably close enough to go home and get it.
Your choice of camp site will be determined by the level of comfort you want, and your budget. Sites with hot showers, toilets, shop and a play area for the kids are great but usually more expensive than basic sites with just cold water and a place to empty the chemical toilet. If you can afford to stay at one of the better-equipped sites, it will give you a slightly more comfortable introduction to camping. When you become tent-hardened, you can then go for more basic sites or even wild camp if you have a light enough tent and equipment. This is free and as basic as it gets, but you get more freedom to choose where to pitch your tent and a real wilderness feeling even when you're pretty close to civilization.
The two main problems that people seem to have when first camping are leaking tents letting the rain in, and being cold at night. Some cheap tents don't have PU-coated flysheets and these can sometimes leak in heavy rain. If you have a tent with a PU-coated flysheet and taped or sealed seams, then leaks shouldn't be a problem. Keeping warm is easier if you have an insulated foam mat under your sleeping bag or bedding. Also make sure the sleeping bag or bedding is thick enough to keep your body heat in, and covering your head will also keep you warmer. Some sleeping bags have a hood, if yours doesn't then consider wearing a fleece hat or similar. Ok, you won't look too attractive in it, but it'll keep you much warmer.