Actually, I'm going to go to a camp just like this next week, so I've definitely got some advice for you! This is my fifth year doing summer camp, so I know what I'm doing!
BAGS AND LUGGAGE:
Normally, I take three bags to camp. Two are trash bags filled with bedding and beach towels, and the other is a large set of storage drawers. Now, I know this sounds crazy but I pack my clothes into a set of plastic drawers before I leave for camp and when I get there I just put the drawers in my cabin. This works so amazing and is incredibly convenient. If you don't have room in your cabin for this or are unable to buy one, just pack your clothes into a large duffel bag to store under your bunk. Another good idea is to bring a couple of milk crates (like the colored plastic ones) to stick under your bed to store stuff in. I've never done this, but a couple of my friends have and they say it works great. You also might want to bring a shower pail for your shower stuff. I couldn't find a link to buy one, but I'm sure you can pick one up at Walmart, Target, Meijer, Kmart, etc. Lastly, bring a small backpack for the daytime to store things like a camera, sunscreen, bug spray, water bottle, etc.
CLOTHES AND SHOES:
Here's what my typical packing list looks like:
6 normal shirts
2 old or baggy shirts to get dirty
4 or 5 pairs of shorts
1 skirt
3 swimsuits
2 pairs of pajamas
7 pairs of underwear
1 swimsuit coverup
2 pairs of flip flops
1 pair of old tennis shoes
1 pair of jeans
1 sweatshirt
I usually bring normal tee shirts to camp, like graphic tees and plain shirts. Then I bring old shirts to get dirty in painting or mud hikes or something like that. I bring a pair or two of Soffe shorts (google them if you've never heard of them, they ROCK!), some jean shorts, and a colored pair of shorts. My jeans are usually old and good for campfires, same goes with the sweatshirt. I bring two pairs of flip flops for just basic activities and tennis shoes for hikes. Lastly, for swimsuits I usually bring two one-pieces and a tankini (I go to a Christian camp, so thing have to be modest). Oh, and also include a couple of camis.
For extras, I would bring maybe another book, some stationary and stamps, lanyard string, friendship bracelet string, an alarm clock, a fan, and some pictures from home to tape on your bunk. Another good idea that my camp uses is bringing a "profile sheet" that you tape to your bunk that's all about you and your likes and dislikes. We also like to bring board games, art supplies, and card decks. My camp has the same "no anything that is electronic policy" and it sucks, but when you bring fun stuff it helps. I wouldn't recommend bringing a digital camera. They can get lost or dirty really easily (trust me, it's happened). Instead bring a couple of disposable cameras so you can have tons of pictures.
Good luck and I hope I helped out!