Here's what I would suggest (and I would go backwards from Japanese101). I bolded some places, so you don't have to read everything.
If you do nothing else, get breakfast at Tsukiji:
Day 1 (western Tokyo): Harajuku, Omotesando, Shibuya, Shinjuku.
Do this on day 2, if day 2 is a Sunday. Start with
Harajuku and spend your morning walking around the
Meiji Shrine. If it's a Sunday, pose for pictures with all the people dressed up. They come other days of the week, but in smaller numbers. Also, the Harajuku train station is pretty cool looking by itself.
Then, walking from the Harajuku station, go down
Omotesando. It has a lot of cool shops. The food there is very Western, but Japanized Western food is interesting. The Harajuku crepe is the local specialty. There are some Japanese curry places and some Omurice (omelette stuffed with rice) places, but there's not much traditionally Japanese. Depending on your price range, it's a decent place for lunch.
After that, I'd go to
Shibuya. You can walk there if you want (there are shops along the way), but the metro will be a lot easier. I think Shibuya is the most vibrant area of Tokyo. It gets called the Times Square of Tokyo. It's a great place for shopping, dining, and drinking. The famous site is the Hachiko statue (of the loyal dog), which is right outside the station. Next to that is the
scramble crossing. Just remember with the scramble crossing, you actually have to get across the street. If you linger, trying to take in the experience, you will get stuck in the middle of one of the busiest intersections in Tokyo. Try to be in Shibuya at night so you can see everything light up. There are plenty of places to eat in Shibuya. Tokyo has so many dining options that it's overwhelming. It's really impossible to tell if the restaurant on the 7th floor is better than the one on the 6th.
From Shibuya, go to nearby
Shinjuku. You can also take these in opposite order, if you want. Shinjuku is one of the major skyscraper districts. The skyscraper area is fairly boring to me, though. Go up the observation tower of the
Tokyo Metropolitan Center to get possibly the best free view of the city (there's a little cafe on the top, too, but I didn't stop). Yodobashi Camera in Shinjuku is a great place to find electronics and cameras, if that's your thing. I say to take Shinjuku last because it has some of the best nightlife in the city.
Kabukicho is a dense collection of restaurants and bars (and people trying to sell you pirated DVDs by day). Near that is Golden Gai, if you're feeling really adventurous. It's a dense collection of hundreds of whole-in-the-wall bars. That's my favorite spot in Tokyo. Memory Lane, in the shadow of the train station, is a great place for late-night eating. The dish of choice is grilled chicken on a stick (Yakitori). There's plenty of beer to go with it.
Personally, I'd stay within walking distance of Shinjuku station and go there. But I don't know how interested you are in late-night drinking/eating.
Day 2 (central Tokyo): Tsukiji, Ginza, Marunouchi, Akihabara or Roppongi.
Get up as early as you can (you'll be jet-lagged, so this won't be too bad) and go to the
Tsukiji fish market. It's the largest fish market in the world and it's actually a place for business, not a tourist trap. However, if you're able to walk about it without getting in the way of the workers, it's really cool. It will only be around in its current form for another two years, so go quickly. If you get up early and get a cab there, you'll be able to see the Tuna Auction. But I hear that's overrated, so I wouldn't bother. But, go to the two famous sushi places nearby (you'll know you've found the right two, because they're basically next to each other and they have long lines with tourists). The set menu is about $35 and it'll be some of the best sushi you've ever had.
From there, you can walk to
Ginza (10 mins tops) to go shopping. If you get a sushi breakfast around 9 or 10, you'll be at Ginza by noon. On some days (Sundays?) they shut down the street to cars so you can walk down it. Ginza's a great place to window shop (or you could actually buy stuff). You could stop to eat there, but it's not cheap.
Marunouchi is another 15 minute walk from Ginza. There are trains that go to all these places if walking isn't your thing. The Imperial Palace and Hibiya are there. I know I slightly dissed the Imperial Palace before, because I find it slightly overrated, but it's still pretty cool. I wish I could go jogging there (as a lot of people do). This area is also one of the most impressive vertically (as in skyscrapers). There is also the controversial Yasukuni Shrine, which honors the war dead who served Imperial Japan (specifically including the war criminals). There's also a war museum, which is essentially nationalist propaganda. If you're into that thing, it's worth it.
If you're into electronics, or anime (or adult videos),
Akihabara is another 20-30 minute walk away. (I really can't blame you if you take the metro at this point). If not, I recommend you head in the other direction to
Roppongi (you have to take the metro there). Roppongi Hills is a nice area (with a nice mall) and you can find places to eat. The big attraction of Roppongi is it is a location for lots of bars and clubs that appeal to foreigners. It's not cheap, but it can be fun (although, I personally hate it there). The Tokyo Tower (another lovely tourist trap) is not far from Roppongi either.
This might be a bit ambitious. If drinking is your thing, Roppongi is a good place to end your night.
Honorable mentions because you don't have nearly enough time:
*Oedo Onsen Monogatari on Odaiba - The hot springs are one of the most relaxing things you can do in Japan. But that'll take half your day. There's too much to see/do in Tokyo.
*Asasuka - This is probably the best shrine in Tokyo. The surrounding neighborhood is pretty cool too. But if you're going to go up here, you may need to sacrifice something else (like the Imperial Palace).
*If you're into cooking, Kappabashi-dori (the kitchen supply district) is a 15 minute walk from Asasuka. If you want Japanese knives (or a coffee roaster), this is the place to go. I bought some cooking chopsticks because I wanted to buy something. Restaurants buy from here, so you're getting the best price available.
*Ueno - It's about a half hour from either Asakusuka or Kappabashi (or you could metro it). It is home to the Tokyo Zoo, the National Museum, the Science Museum, and the Western Art Museum. Asasuka and Ueno is a day.
*Shimokitazawa - There's almost no chance you're going here, but it's the quirkiest, most hipster-friendly area of Tokyo. It's all used-clothing and obscure toys stores. It's a hangout for students and people in bands. There's also a coffeeshop where the proprietor makes one of the best espressos in the world, but only the morning, because he won't make it after 3-ish because the quality declines (he'll make cappucinos and lattes, though).
Here's where to go if you're looking for specific Japanese foods:
*Sushi - Tsukiji. Unless you're rich and can afford Gizna, accept no substitutes. Daiwa or Sushidai are both fine.
*Yakitori - Memory Lane, as mentioned earlier. Yakitori is all about atmosphere and this has tons of it.
*Ramen - This one has some options, so I'll go into detail:
1) For authentic, Tokyo-style Shoyu Ramen, go to Harukiya in Ogikubo and order the Wonton-men. Fish broth and soy sauce give it its unique flavor. It's out of the way (15 mins west of Shinjuku by train), but worth it.
2) For Hakkata (aka Fukuoka, Kyushu)-style Tonkotsu Ramen, go to Ippudo in Ebisu (they have other locations. They have a branch in New York, where everything is double the price. Tonkotsu Ramen is incredible (tons of pork fat).
3) For Ramen that is basically a cult, go to Ramen Jiro (I've been to Ikebukuro). There are 28 locations and they're all a little different. Get extra garlic, but don't get extra veggies or meat. The thing is huge, so extra anything is overkill.
These are places I've been to and can recommend. All Ramen places in Japan let you order by buying a ticket at a vending machine, so it's not too bad for those who lack Japanese. Here's a random link of ramen places, that'll substitute for me naming places near the major locations:
10 ramen shops in Tokyo worth visiting | Matador Network
Two Japanese dishes that you won't find in great quality in Tokyo are Takoyaki (octopus dumplings) and Okonomiyaki (meat-veggie pancake).