I think for the most part these guys summed it up pretty good.
I do firefighting in the summer and have to carry all my gear with me and that has transfered over to my stuff I use for camping, trust me, lighter is better!!!!
Start with your shoes, find something that is lightweight, very strong, and most importantly comfortable. Wear the shoes a lot so your feet are use to them and so that you know you won't have problems with them when you are away from home, I have had awesome luck with Merrell(sp?). Good socks is also a must, I use a sock that I bought at Sportsman Warehouse that is like an army sock, nice and think but wicks sweat and keeps your feet cool. You want somehting like that so you don't develop blisters, heat + friction = blisters.
As for light and small, I use a northface tadpole 23 tent, weighs absolutely nothing and is strong as heck. Got caught in a windstorm/thunderstorm last summer and slept like a baby, no worries. (buddy got blown down the beach in his tent, different story though lol) Sleeping bags you can definately find small ones, mine is maybe 6" x 6" round and maybe 10" long, and is very warm for summer camping. Like someone else said, buy according to your core temp at night, I sleep in a t shirt and my shorts with a beenie on, stay plenty warm.
Splerge and buy some of the techy camping gear, get a jet-boil, a good filter, some LED style lantern/flashlights, a good torch style lighter, alcohol based hand sanitizer, etc. and will make the experience tha much better, less you need to worry about going wrong the better.
Food I am not a picky eater, bring some jerky, mixed nuts, etc. My favorite is spam when camping, open the top, add a small packet of koolaid style drink mix to the top, throw in fire till cooked, tastes great!!! MRE's are easy to go though also, see if you can't find a deal on them in bulk, at elast you know you will be getting all your nutrients you need.
Clothes, almost forgot, buy some of those convertible pants where they zip off to shorts, very nice to have with temp changes, the pair I have are almos like swimsuit material, dry really quick but a lot stronger than typical nylon. I always have a tshirt as a base layer (underarmor tshirt material), a light base Northface fleece in case it gets a little cold, and a northface windstopper coat that I pack with. (another trick is buy a good rainsuit, some you can close the vents that breathe on them, warm as heck to put on as a middle layer...)
Need anything else before you go buy just post it up, or PM me