If you like to try sushi, then you should go to "Tsukiji shijo"
It's a fish market where you can get a nice and fresh seafood or fish meal.
How about shaved ice (kakigori) ? Recently, there are many shaved ice specialized restaurants have opened, such as Yelo in Roppongi, and ICE MONSTER in Omotesando. Himitsudo in Yanesen area is a long established shaved ice cafe, where hundreds of people line up during summer!
I recommend Yakitori! It’s a Japanese style grilled chicken. You can find yakitori in most Izakaya. It goes with both beer and sake. There are many kinds, made by different parts of chicken body, so try variety of Yakitori yourself!
One of the most famous Tokyo local cuisine is "monjayaki".
Monjayaki is known as another from of Okonomiyaki, but it has different ingredients and broth. It requires instructions to make, but making a monjayaki is also a fun thing to experience! Tsukishima district near Tsukiji is especially famous for monjayaki. There is a street called "Monja Street", with over 100 monjayaki restaurants.
If you like sweets, I'd recommend trying ningyoyaki. You'll find ningyoyaki easily at Nakamise-dori, a street you will encounter when you visit Sensoji in Asakusa. It's a doll shaped pancake with red bean fillings.