Kyoto is a must see for every visitor to Japan. It used to be the capital of Japan for over 1,000 years! There are nearly 2,000 palaces (split between Buddhist temples and Shinto shrines), lots of castles, and more importantly some of the best food in Japan.
When you travel to Kyoto it is almost like a time machine that takes you way back. It was nearly perfectly preserved during WWII and remains one of the oldest places I have ever visited.
To get there, I would recommend taking the Shinkansen (bullet train) from Tokyo and getting there in about two and a half hours.
In my opinion the best time to go is in the spring. I think the temperature is the best and if you get there with the cherry blossoms, you will not be disappointed.
The best food by far is takenoko, which is an edible bamboo shoot. I know this sounds odd- but trust me on it. I frequently request extra takenoko in my ramen or pickled.
Give it a shot!