Given your age and interests, absolutely Oahu is where the action is at. It has a little bit of each of the islands (except of course the Big Island which has an active volcano!). More specifically, you will want the excitement of Waikiki if you want to see a lot of other young people from around the world. Some of the other resorts off Waikiki cater to older people. That rules out the Kahala Resort, Turtle Bay and some others. I don't think you will enjoy KoOlina which is far from the excitement of town.
I belong to CostCo, looked at the available hotels. The cheapest Waikiki hotel rate listed, Aston Waikiki Beach Hotel is a decent hotel; not a luxury hotel, but clean and in a great part of Waikiki. The staff is nice. Note that all hotels in this part of Waikiki are directly across the street (Kalakaua Ave) from the beach and not right on it, but it really doesn't matter much. You will have more fun on this side of Waikiki, near the International Marketplace and a lot of surfing rentals, the zoo, aquarium, and Kapiolani Park. If you can afford it, the Moana Surfrider is a cool historic hotel, the last one directly on the beach in this area.
If you want to be in a self-contained mega-resort, then I like the Hilton Hawaiian Village, but this is on the opposite end of Waikiki. The Waikiki Sheraton is in a string of hotels owned by Sheraton right on the beach in the middle of Waikiki, but I don't think it is designed as well as the Hilton, and always seems congested to me.
Absolutely rent a car so you can drive around the island, go to North Shore to visit the surfing town of Haleiwa, see Sunset Beach, go to different snorkeling places (Hanauma Bay, Shark's Cove), different isolated beachs (Kailua, Lanikai), go hiking (Manoa Falls), sightseeing (Pearl Harbor, Iolani Palace, Polynesian Cultural Center) and so forth.
Oahu should be your first island, but why not visit more than one island? Unless you expect to sit on the beach all day, I would think you will be bored after 4-5 days on any island, so if you are staying more than a week, absolutely consider jumping to another one. Cost will be about $80-$100 extra per person for this luxury, but worth it if you can swing it. Remember rental car rates will be slightly more as a daily rate vs. a weekly rate however.
If you want the perfect American (less international) tropical resort away from urban environment, go to Maui. They have a new cool activity call the "zipline" there where you jump off cliffs on a line, you can drive the windy cliff road through the jungle to Hana, snorkel with sea turtles, ride a bike downhill down a volcano.
If you want isolation and the "beauty of the South Seas" that is Kauai, but don't go here if you are easily bored.
If you want the most exotic island, that is the Big Island, but it is immense, and driving distances are far, so you have to like to drive. The up side is seeing an active volcano, swimming with Manta Rays, walking through a tree fern forest, hiking Akaka Falls near Hilo. The negative is that the island is too new to have much coral and lagoons, so there are very few beaches here!
Have fun!