I've been to a few of these places and have been around the world a couple of times (literally, on a US Navy Guided Missile Cruiser), so perhaps I have some insight. (But then, I was a drunken, single (with all that that implies) sailor. You may not want to do some of the same things that I did in my youth. The other caveat is that I did all my traveling 15-20 years ago, so some things may have changed.
Africa - I went to Kenya and went on a photo-Safari for three days. Best three days of my life. It was absolutely superb. Go there. Go see the place. Go see Mount Kilimanjaro. Go see cheetahs and elephants and gazelles and all the rest of it not locked up in a zoo somewhere. A short google searched turned up this. That's close to what I did. Salt Lick Lodge is an amazingly cool place. You've got to see it.
Hong Kong - whatever else you do, take the hydrofoil over to Macau and stroll around. Lots of interesting things to see. And gambling. If I remember correctly, Kowloon had all the best bars and shopping.
Australia - I only ever went to Perth. Friendliest bunch of people you ever wanted to get drunk with. They made us wear our uniforms off the boat the first day. I didn't buy a single drink all day.
Philippines- I'll let Oishi be your guide there. I never went further than a mile from the gate outside the naval base, anyway.
America:
New York - obvious. Must do.
San Francisco - good tourist town. You've got to go to the 'Buena Vista' and have an Irish Coffee. It was invented there. Almost no-place else in the world makes one as good. Also, in my opinion, North Beach Pizza makes the best pizza on the face of the planet. At least they used to. If you can figure out how, try to see Yosemite National Park. One of the most beautiful places on the face of the planet. Achingly beautiful. Pictures don't do it justice. Only a couple of hours from S.F.
LA - not recommended. Unless you're star-struck by Hollywood, there really isn't much to do. I guess you could stroll Rodeo Drive. Or take a tour of one of the studios. Meh.
Las Vegas - I'm going there on Sunday. Not sure what there is to do there. I'll let you know when I get back. :P If you're going anywhere near there, though, you've absolutely got to see the Grand Canyon. Rent a car. Take a tour. Whatever you need to do, but go see it. You'll regret it if you don't.
Might I add Washington, DC? If nothing else, for the Smithsonian. Air and Space; Natural History; American History; Gallery of Art; that's the big 4. Hard to see them all in one day. The newly opened WWII memorial is pretty good. I've also got to put in a word for the Museum of African and Asian Art. Not so much the African, but the Asian stuff is way, way cool. Samurai armor. Edo period prints. Vases. Weapons. Did I say it was cool? Also, the Gallery of Art has an east and west wing. The west wing is the traditional collection (Monet, Rembrandt, Botticelli, etc). The east wing has rotating temporary exhibits which can be really awesome. They had an exhibit of Olmec artifacts. The had one on 16th century Portuguese baroque art. (There was a horse-drawn coach there that was the gaudiest thing you could ever imagine. I think the word gaudy was invented to describe it.)
I know Europe has really good art galleries but the Smithsonian is not bad. You should give it a try. Best thing: It's all free. Not a penny. Nothing. You've got to walk through metal-detectors these days, but it's still free. Plus, the Lincoln Memorial is pretty impressive. Try to come in the off-season. (In summer, school's out and everyone brings their kids.) (I would offer you a place to stay if I was single, but my wife would probably object to "a pretty young single woman friend I met on the internet".)(I actually hosted a couple of Brits once a long time ago. They wanted to see the American History museum. They said they could see all that other stuff in other countries, but who else would know American history best?)
Brazil - Carnival is the greatest party on the face of the planet. The beaches in Rio are some of the best in the world. They have great coffee and really good steaks. And Brazilians know how to live. General day: Get up at 7:00. Go to work. Drink a little at lunch. Knock off work at 4:00. Get some drinks with friends. Go home, take a nap 'til about 10:00. Get up, freshen up and head out to the clubs. Party until 3:00 am or so, then go home and get a couple of hours of sleep before starting the next day. And they do this continuously. Non-stop. Every weekday of the year. Don't ask me about the weekends. I assume they sleep, but then I saw plenty of them running around on Saturdays and Sundays, so I'm not sure. (And some of the most beautiful women on the planet hang out there on the beaches. Not that you'd necessarily care about that. Just an fyi.)
I second Todd's suggestion of the Caribbean and seeing Mayan temples. If you could work Hawaii in, you probably should, although I doubt it could beat Fiji. Might be a cheaper alternative, though. Although if you're in New Zealand, Fiji might be cheaper to get to.
Might I also throw in a suggestion for Thailand?
The Temple of the Emerald Buddha. Also, they have these 'World Bars'. Imagine the biggest warehouse you've ever seen except taller. Filled with people and giant video screens. Huge columns with walkways and terraces. I don't know if they still have them or if places like that are commonplace in most countries now. But I still haven't seen anything like them anywhere in the US. So maybe not. Plus, the food is awesome.
That's about all I can think of.
