Over the years, various friends have asked me for suggestions of fun things to do in Tokyo during a visit. This list, which still changes, is what I have built as an answer and it’s targeted for people who don’t speak Japanese but are visiting Tokyo for some short period of time. This list isn’t meant to be some snobby locavore view of Tokyo. This is stuff that I think my friends would enjoy doing.

Metropolis is a free English-language weekly that covers what's happening in Tokyo. There’s also Time Out Tokyo. Both offer schedules of weekly music, arts, sports, etc. events.

Ippudo ramen is a chain (and not vegetarian at all) but delicious.

At some point, you should try kaiten-zushi -- the sushi on conveyer belts.  Restaurants of this type are everywhere and you can easily duck in. (Side note: if you don't like mayo, be willing to say that a lot. You'll be very surprised by how much mayo Japanese cooks use, even / especially on sushi).  Here’s one restaurant I like, Bintokona, despite the fact that it’s in Roppongi Hills, a kind of embarrassing mall.

Tsukiji fish market is a fun visit, especially early in your trip when you’re jet lagged and therefore have no problem waking up at 3am.

Comme ca Cafe for fruit tarts.

Dengama has great, not-too-expensive china and dishes, and it’s in a neighborhood, Kappabashi, full of kitchen goods shops.

All pens and stationary in Japan are more amazing than everywhere else, but I like the first floor of the Loft in Shibuya.

Any department store is fun to explore but the basement level is an awesome display of food. Takeshimaya is my favorite but they’re all pretty similar.

I know Grom is Italian but I still put it on this list. Right now, the Tokyo branch is on the first floor of the main Maru-ichi building in Shinjuku.

/ Theater
The Edo-Tokyo museum, if you want a history of Tokyo. It’s packed with exhibits and will mean you get over to the other side of Tokyo, which could be nice.  Within walking distance of the museum’s north side is another fun area around Asakusa.

The National Theater with Noh and Kyogen, depending on the schedule. Kabuki and Bunraku (my favorite) are here.

Walking / Neighborhoods
You could easily spend your time in all the big, flashy parts of Tokyo but that would give a very warped sense of the city. By all means check out Shinjuku (Kabuki-cho is the especially trashy bit), Shibuya (Center street), and Roppongi (full of foreigners). But I recommend making a point to get out of those areas, too.

There are lots of great neighborhoods with clusters of particular goods, including: used books in Kanda, musical instruments next to sporting goods stores in Ochanomizu, electronics in Akihabara, quant old European-ness in Kagurazaka. Harajuku has become made more famous recently (to non-Japanese especially) as the trendy street fashion place.  A popular street fashion magazine called Fruits regularly stops people on the street in this neighborhood and asks them to pose for cameras.  Near Harajuku, there is also an entrance to Yoyogi park, which has 1) crazy street fashion and great self-conscious youth sub-culture gatherings on the weekends and 2) Meiji shrine, which is a lovely green oasis in the middle of the city. Shin-Okubo is Korea-town and has lots of great food.

Ueno is one of the most famous park areas in Tokyo, and home to a very famous zoo (that was a much bigger deal before and during WWII).  Has a bunch of art and history museums in the park, as well as the zoo (which I've never gone to).  Ueno is also home to a large number of homeless people, mostly men. Some tourists mistakenly believe that poverty or homeless aren’t problems in Japan.

Day Trips from Tokyo
Yokohama is always fun, especially if you want to see a Japanese Chinatown.

Kamakura isn’t too far and has a giant buddha you can climb in.

You could go visit Mt. Fuji or try to climb the mountain.

Preparatory Reading
Some people have asked me for reading suggestions. The best suggestions I’ve got are the readings in my introductory course Japanese Culture. You can pick and choose what looks interesting.