They have fish and beef and pork. They have rice and noodles. They have doughnuts and French bread. They have ice cream stores. They have green tea and cappuccino. They have burgers and pizza.
To summarise, Japanese food isn't just rice and raw fish.
In Tokyo, my favourite place is to eat is Chikara Meshi. You order from a vending machine (a total win for those with little/no Japanese), take a seat at the counter and get served quickly. It's also super cheap and delicious, and usually open 24 hours. They have different menu options depending on the location, but they all seem to have the standard curry rice and "gyudon" which is a bowl of rice topped with strips of fried beef or pork. . .It is super super yummy! A normal size of one of these will cost you 290yen, which is really cheap.
Tonkatsu is another great Japanese fried food that everyone should be made eat all the time. There's a chain called Katsu-ya that are quite cheap (700yen), and another called "Wako Tonkatsu" which is more expensive (1100 yen) but you get free extra servings of rice, miso soup and cabbage.
I've also come across Yayoiken, I'm not sure if it is a big chain store but it has a great selection at affordable prices (700-900yen), and also has free extra rice servings.
Tonkatsu! Behold the meal of Kings. . or is that Emperors?
If you want to try a Japanese burger, the natural option is Mos Burger. It's pretty good chain store and not too expensive, with a burger, fries and drink costing about 700yen.
There's McDonalds, which has the usual you expect plus lots of yummy extras that come and go with the season. I'm not sure about America, but that only happens very rarely in Ireland, while Japan seems to have a new menu item or special offer on every week. Burger King is the same kind of deal, but Denny's has a totally different Japanese style menu, so be warned! There are some Subway restaurants, but they weren't so common. I think the menu is more Japanese-orientated and therefore probably differs from the usual options, but you should probably check it out yourself!
The only McDonalds sign I ever saw in Japanese!
An 'ebi' burger (that's shrimp!) It was a bit bland but otherwise okay
As for coffee and desserts, there's Mister Donut which is, as the name suggests, a doughnut shop. They all cost around 120 yen, and often have sporadic 100 yen sales. There's also many Vie De France bread shops which have savoury and sweet breads for around 150yen. The most common coffee shop I see is Doutor, but there's also plenty of a place called Becks, and the obligatory Starbucks. Baskin Robbins ice cream shops are quite common, but they're usually referred to as "31 Ice" in Japanese (because of the 31 flavours).
Mmmmmmm
Spicy chicken balls and rice from 7-Eleven
If you don't speak Japanese it can be kinda scary getting food, but apart from the aforementioned vending machines, they are also displays in some restaurant windows. Then you can pretty much see a decorative selection of their food, so you can get a good idea of what's available without having a word of the language.
Plastic model pancake display in cafe window
You may have already heard, but vending machines are still really popular in Japan. In Ireland they're a dying commodity, but in Japan you can find at least two on every street corner. They are literally everywhere! However, the vast majority of the time they only sell drinks. I think I only ever saw one with food in it, and even then it looked like a breakfast bar kind of thing. There are also rumours you can get vending machines with weird stuff in it (like underwear. . .) but I can assure you I never came across any of those. You can however get hot drinks like green tea and coffee in these vending machines, which is kinda fun! Prices are around 120-250 yen.
The Japanese are (from what I've encountered) really eager to help you, and while they will usually not have the best English, they're going to really try and understand what's going on. So I say whip out your Japanese phrase book, point and gesture, be patient, and everything will run relatively smoothly!
As for a disclaimer, I live in Tokyo and this has only been my own small experience. Somewhere like Tokyo obviously has a lot of Western food options because it is a big city with a large amount of foreign tourists and a lot of business ties with America, if you go to smaller areas or towns I imagine your options would be more limited to traditional Japanese food.
A small selection of pot noodles from the museum in Yokohama