If I spend 5-6 days in Tokyo, is it easy (transportation-wise) and worth it to take a day trip to Kyoto, or is it better to just focus on exploring the admittedly huge city of Tokyo?
Kyoto is a remarkable city, I totally loved it, but for such a short period of time is not worth it, because you'll miss everything important. The train (which is pretty easy to take, but not cheap: +$150 one way) takes around 3 hours and once in Kyoto you'll only have time to go to one area and come back to the station.
There are so many things to do in Tokyo that 5 days is not enough. Yesterday I wrote this, actually, to help you plan your first trip to Tokyo: http://hayo.co/how-to-plan-trip-to-tokyo-basic-beginners-guide/
Have fun, Tokyo is fantastic!
Tokyo and Kyoto are in complete contrast to each other. You can enjoy the sophisticated pleasures of Japanese city life in Tokyo. In Kyoto, on the other hand, you can enjoy traditional Japanese culture such as temples, tea ceremony, Kimono etc. If you want to understand and feel the spirit of Japan, I recommend you to take a day trip to Kyoto. It takes about 2.5 hours by JR Shinkansen (high-speed railway lines in Japan) from Tokyo to Kyoto.
I have spent 9 days in tokyo and they were not enough! Every district is massive and unique that you would want to experience all of them.
kyoto seems like a small city compared to tokyo but it is not! Every temple in the surrounding hills is worth visiting. I did spend 6 days there and were maybe a bit too much, but a day trip will not do...
have fun japan is a unique experience.
Here's two posts I wrote to give you an idea about the best things to do in Tokyo and some advice on the costs involved.
So, people here are clearly divided, but I think it would be a real shame for you to miss out on Kyoto. Yes, Tokyo is huge, dense, varied, and you could never run out of things to do there. But it's really a hectic place, and Kyoto moves at a very different pace. Going there will let you see another aspect of Japan that you don't get in Tokyo. And an evening stroll down the street called Ponto-Cho will be incredibly memorable for you. Yes, it's a 2.5 hour train trip on the bullet train, but the train is actually really fun and comfortable. And Japanese train stations are super fun, too. I love the Kyoto station. There is a ton to do and see just there. Just doing Tokyo would be limiting yourself and will be more one-note.
Mentioned in this answer: