Did a similar trip in my medical elective a few years ago, you'll have a great time.
Look in to hiring a car in the states if you can, I can't remember if I hired a car younger than 21 in the states but some companies (including Budget) will allow you to hire if you pay a daily supplement (something like $25/day)... There's nothing like being out on the open road in California, especially in a convertible

One time we used a company called usrentacar.co.uk - looking at their website it says you have to be 21 but they might be able to help/give advice.
I'd recommend checking out Hollywood, Sunset Blvd, Melrose Dr, Santa Monica, San Diego (awesome night life and young crowd) and take a trip out to Las Vegas/Grand Canyon if you can! San Francisco is awesome too if you can make it up north a bit.. Make sure you try an In-and-out burger LOL. You're spoilt for choices with beaches, one of my favourites was Mission Beach in San Diego...
New York - the usual things, Empire State building, statue of Liberty, take a ferry tour, Wall Street, check out the Park and Bryant Park... Walk around, Times Square, Broadway, Macys, check out the camera stores if you're in to that, B&H etc
Hong Kong - so much to do here, I'd pay a visit to Dragon-i (night club) in Lan Kwai Fong - the girls here are amazing here especially on a models night.. Check out the Peak, Big Buddha and cable car (worlds longest). Mong Kok and Sham Shui Po for electronics... I'd recommend a cocktail at Felix at the Peninsula hotel (one of worlds top rated hotels, has a fleet of RR Phantoms).. Also check out Azure bar in Lan Kwai Fong for a good view. If you fancy a good steak check out Morton's steak house in the Sheraton... Take a boat trip - on the Star Ferry at the very least, you can hire junk's I think or find someone with a private yacht at Dragon-i! Stanley Market for the odd clothes bargain.. Temple Street market for the experience and tat... If you're in to electronics then check out the Sham Shui Po street markets, they sell EVERYTHING here.. Check out some of the malls, IFC is probably the best one these days. Disney land for about $35 a day if you're in to that kinda thing, or Ocean Park is a good day out... Car tuning shops in Mong Kok.. If you're really in to cars then you could check out one of the car supermarkets, there's one under the Wan Chai convention centre which has all sorts from Ferrari's, Lambo's to Nissan GT R35s etc.. Clearwater Bay and Repulse Bay for beaches (near Stanley)... Maybe go to a horse race at the Jockey Club if you can arrange it. Don't be afraid to use taxi's, they're really cheap, $15HKD starting fare for the first 2km...Otherwise the MTR (tube) and bus services are great and very cheap. Take a tram whilst you're there too...
You could also think about taking a day trip from HK to Macau, costs about $150HKD return on the turbo jet boat, things to see and do there, and might give you a taste for Vegas - check out the new massive Venetian they've built there. You can also go go-karting here, good Portugese food and interesting night life...
New Zealand - I spent a week driving around the north island. Go to Rotarua, do some of the outdoor activities, black water rafting, louge, etc. Pay a visit to the hot thermal spas. If you're a skier take a trip to Whakapapa for a day. Go up to the bay of islands and swim with dolphins and just generally enjoy the beautiful scenery. Don't stay too long in Auckland, not a huge amount going on here.
On my trip I rarely booked accommodation in advance to give me flexibility. This was absolutely no problem in the states, a multitude of motels with availability and open late. NZ things are a little quieter, but if you can ring ahead you'll be fine armed with a guide book.
Like you say it'll be expensive. If you don't mind hostels things will be a fair bit cheaper but enjoy it and worry about the money when you get back
Cheers,
Francis
Just to add, from past experience, you can stay in motels in California for around $50-80 per night in general for a double room, Motel6 are even cheaper I think but don't always have availability. We stayed at the Vagabond Inn in Hollywood for around $100/night, nice and central although some dodgy night-time action going on there! Hostels wise, my brother said the "Big Apple" in NYC was good, I stayed at a few Hostelling International (
http://www.hihostels.com/) ones in SF/SD which seemed decent but probably close to $30/night and I soon got tired of dorms..
Shame about the current exchange rate... I just got back from a road trip around France, now that place is extortionate at the moment!