Osaka! When you go there, keep singing the Shonen Knife song.
"I have been to London one time. I have been to Hong Kong one time. But my favorite town's Osaka! My favorite town's Osaka!" I haven't been to the Kansai area for such a long time, but I still consider myself a Kansai boy.
Shinsaibashi! Take the Midosuji subway line to get there. Start out at America Mura. It's a great place to go if you're hungry for.... takoyaki. Yeah, not a decent burger joint in site, myseriously. All they have there is a McDonalds. Ame Mura is interesting though. Guys from Africa pretending to be Americans, selling tennis shoes and such. Lots of interesting American toys and collectibles, many of which you can't even find in American shops. Mike's Toys is fun. Walk down to the Dotombori canal. So fun! Eat okonomiyaki, see the giant robotic crab, people watch along the canal. Actually, the canal is beautiful at night. Get your picture taken in front of the giant Glico marathon running guy mascot if you like Pocky sticks. There's an interesting underground shopping mall in the area called Namba Walk. From the Namba area, and from there you can make your way to Den Den Town, te Osaka equivalent to Akihabara.
DEFINITELY see the Osaka Kaiyukan Aquarium that UD mentioned. I think the only more impressive aquarium in the world is in Spain, from what I've seen on TV. Don't try to do the Aquarium the same day as the other stuff I mentioned above. Oh yeah, and there's also a Mandarake store near Osaka station/Umeda Station.
Go to Kobe if you're staying in Osaka for any number of days. It's a quick ride by taking the shinkaisoku there. Get off at Sannomiya station, head a block or two south until you see the big shotengai (covered shopping arcade). Follow this until you arrive in the Motomachi area. Here is Kobe Chinatown, yay! Find a street vendor selling Kobe beef on skewers for TEH NOMZ. The series of shotengai continue all the way to Kobe Station area. If you head back to the JR track, you'll see a neon-lit sign saying MOTOKO. Motoko is a cramped, narrow shopping hall extending along underneath the JR line. It's FULL of junk shops. Junk electronics, junk clothes, junk trinkets, junk antiques. There's a retro game shop in there too (I hope it's still there). It's fun to poke around down there. You can find boxes of Famicom carts that have been dumped there, recycled clothing from America, etc. Kobe station area is real fun. Head south to Harbor Land! It's beautiful at night. You'll see the Mosaic building where there are lots of neat shops, a Snoopy store, gelatto, a small amusement park with a beautiful ferris wheel, etc. Short boat cruises from there can take you to see the Akashi Bridge, the #1 longest suspension bridge in the WORLD, which is also beautifully lit up at night, and the colored lights change depending on the season. It connects the city of Akashi to Awaji island, which is between Honshu and Shikoku and was the epicenter of the great Hanshin earthquake in '95.
You may not find much on Kobe in Lonely Planet or other travel books, but just make your way to Sannomiya Station there and you'll find an information booth that has free maps of the area. It's real hard to get lost there, since everything either goes along the tracks or heads south to the ocean. Just go and you won't regret it. Kobe's my favorite city in Japan, #2 being Nagasaki.
I wrote this stuff about Osaka and Kyoto from my first trip to Japan in '98. It's old, but it has some useful information.
Oh yeah, and it's Odaiba and Harajuku. You misspelled those words.
