Meiji Shrine, Asakusa Kannon Temple & Nakamise Shopping Arcade, Imperial Palace East Garden.
Not so obvious things:
Ueno Park
This has to be the best park in Tokyo, with the Toshogu Shrine, Tokyo National Museum, The National Museum of Western Art, Shinobazu Pond (a bird sanctuary), noodle stalls, bars and cafés. Many local families come at the weekend, and it's a lovely place to stroll. One of the main places to see the cherry blossom in spring too.
Ameyoko Wholesale Market
An interesting street market with stalls selling all kinds of things. Certainly worth a stroll. Stretches from JR Ueno station to Okachimachi.
Shinjuku
The area around the corner from massive Shinjuku station is fascinating: small streets crammed with noodle eateries and other shops.
The area of 2-chome is allegedly the gay district of Tokyo (see Time Out guide), and Kabakicho is the red light district.
Shinjuku Gyoen (Shinjuku Park) is a haven of peace, surprisingly large, another place for seeing cherry blossom in spring.
One of the biggest buildings in this district of skyscrapers is Tokyo City Hall: for a panoramic view of Tokyo, go up to the viewing observatories on the 45th floor of either tower (especially good on a good sunset, or after dusk). Open until 11pm, and free.
Ginza
One of the most famous (expensive) shopping areas of Tokyo.
Hakuhinkan Toy Park is a great toy shop.
Visit the Sony Building to see (and play with) the company's latest products.
Toky Kyukyodo is an intersting shop for tradition crafts, on Ginza dori a few shops away from the Harumi dori intersection.
For a taste of the Ginza, see this presentation.
Shibuya
A trendy area for young people to shop and hang out. Many small cheap restaurants.
The TEPCO Electric Energy Museum has eigth floors of fascinating electrical gadgets (open 10:00-18:00, closed Weds)
Things not to do:
Tokyo Tower - although it's based on the Eiffel Tower, somehow it's unimpressive and tacky. Not worth a special trip.