Hi, George
Firstly, I hope your dates are not 4/29 - 5/5. That is Golden Week in Japan. A major holiday week that tourists should avoid.
As you like food, I suggest you buy Yukari Sakamoto's Food, Sake Tokyo from Amazon. Not only will she give you an intro to Japanese food, she breaks down Tokyo into neighborhoods with maps of restaurants and food shops. This would be a good guide for you. Also, she includes the Japanese characters (kanji) so you can just show that to the Japanese for directions.
For Kyoto, you might think about staying at the temples which are fairly inexpensive and would give you "a real feel" for Japan. Try myoshinji.or.jp, monbou.jp, Ninna-ji, or Shunko-in.
Another option would be to stay near Kyoto Station, then you can go to the station and eat. The Isetan department store on the 11th floor has some very fine restaurants: Tenichi for tempura for about $40, ask for a seat at the counter. The chef will fry up your food and place it directly onto your plate. How fresh is that?!. Wakuden offers modernish Japanese cuisine but very good, about $60 and up depending on what you chose. Again sit at the counter, you will be facing the window and see the Kyoto night light. Restaurants there that we didn't get to but on our wishlist were: an all tofu restaurant, Kyohyakusai-a buffet of Kyoto home-style cooking. I recommend going here because the staff speaks English and you will not have any problems making reservations. These places are busy. Chairs are lined up outside for people to sit and wait.
On the 10th floor are a series of ramen noodle shops from various parts of Japan. There is a similar one at Tokyo Station too.
At Kyoto Station, outside by the taxi stand, go downstairs to the Porta, another shopping complex. There is a restaurant that serves okonomiyaki - sort of a pancake/omelet, unique to Osaka/Kyoto area. A must eat. Inexpensive and good. You can get okonomiyaki at other restaurants too.
If you go to Nijo Castle, then have lunch at Muten Kura Sushi, which is just two blocks west of Nijo Station. This is a fun conveyor belt sushi place at 100 yen/plate. You can google YouTube to see how it is like.
If you go to Gion, across from the Yasaka Shrine is Izuju which serves Kyoto style sushi. What we know as sushi is Edo style (Tokyo style). Kyoto is not close to water, so its sushi has more vinegar for preservation. Not everyone's taste, but worth a try.
If you go to Arashiyama, there is a one Michelin star grilled eel restaurant. Or, you could have a Buddhist temple meal at Tenryu-ji.
There are lots of Japanese chain restaurants that are very good: MosBurger - order the beef rice burger. Skip the fries, nothing special. CoCo ichibanya - Japanese version of curry, nothing like the Indian. Yoshi Beef Bowl - very tasty and inexpensive.
For something fun and not touristy, try taking a Taiko drum lesson either in Tokyo or Kyoto. There is an English website to register.
Of course there is Tsukiji Market in Tokyo and Nishiki Market in Kyoto for more food.
Personally, I would leave Tokyo as my "big" city experience and minimize the time in Osaka. You can take some day trips out of Tokyo and/or Kyoto. Or go to Hakone on your way to Kyoto. Stay at a ryokan and get to soak in an onsen.
For more food information, look at YouTube: Trails to Tsukiji, Begin Japanology, and Core Kyoto.
Also look at Tokyo and Kyoto forums on TripAdvisor.com
Have fun.