Kawagoe is located in Saitama prefecture, which is right next to Tokyo. There you will be able to encounter traditonal Japanese style buildings. Also there are shopping streets to buy some cute souvenirs.
One of the most easy place to access to experience local & traditional atmosphere of Tokyo. Going to temples and having lunch is also a fun, but I'd recommend trying Japanese desserts such as odango & anmitsu. There are lots of Japanese cafes in Asakusa.
I'm not sure if Akihabara is a place to experience Japanese culture you are expecting... but I still think Akihabara is a fun place with unique characteristics. This may depend on your interests.
Try staying at ryokan, traditional Japanese inns. You can expect lots of Japanese things such as onsen and kaiseki cuisine. These are not Tokyo but old building districts and traditional cultures are lot more in Kyoto than Tokyo.
The Imperial Palace is a beautiful place to see and even just walking around the palace , you will see many famous spots, like Tokyo Tower, The National Diet Building and Yasukuni Shrine. If you have a chance, cherry blossom season is the best for this area!!