A few things to keep in mind......
If you change planes and fly to Tokyo from Los Angeles, most of those flights arrive in the late afternoon. The next day. Example....your flight leaves the U.S. on Monday. You will arrive on Tuesday. You cross the international date line. On the return, you will actually arrive before you left. (Keep this in mind when you make hotel reservations..)
Because those flights arrive in the late afternoon, by the time you get to your room and get something to eat, it will be time for bed and a good night sleep. This may help alleviate jet lag.
If you want a leisurely first day, and you do stay near the Tokyo Station, I suggest you wander around the Ginza district which is within walking distance. This is the upscale shopping area. However, there are a few large department stores with amazing food halls. Some have an entire floor of just confectionary. I enjoyed wandering around and seeing the different foods.
In Akihabra there are a few department stores dedicated to electronics. I'm talking big multi-story department stores. The choices of products is unbelievable. As an example, electric shavers are popular in Japan. At one store I visited, there were about 100 different shavers on display. I don't now how they choose.
If you run across Daiso, step in and browse. They are all over the place and are Japan's version of a dollar store. (100 yen store.) Only it puts our dollar stores to shame.
In Kyoto, I stayed at the hotel inside the actual train station. The Granvia. Very nice but not cheap.
The Granvia had an amazing, huge dinner buffet with both Japanese and Western foods. I don't remember the cost.
The train station itself is a place to visit. There are two shopping malls with over 100 stores and at least 50 places to eat. There is also a huge department store with a set of stairs--and escalators--that go from the bottom to the top. All outdoors. At the top is an observation deck with great views of Kyoto. Also on the top floors are restaurants. I remember one area had five different ramen shops all specializing in a different type of ramen.
The store is called Isetan:
I don't see Sanjusangen-do on your itinerary. It's something you should see. http://sanjusangendo.jp/
If you do find you have extra time, which I doubt, you could take the train up to Nara. (45 minutes by train from Kyoto.)
Lastly, the bullet train is called the Shinkansen. Trying to buy a ticket from the US is a royal pain. Unless you are there on a holiday, you shouldn't have trouble getting tickets once in Japan.
Another lastly, Japan is very safe and serious crime is rare. There is also no tipping.