The Tokyo subway system might look intimidating, but it's very efficient and easy to use. The trick is taking a deep breath and finding the English map.
When you approach the subway station, you will see the lines displayed above the machines from which you buy your tickets. What you need to do is count the stops you need to go. There will be a number circled above the stop. This is the fare required to travel to that station. For example, if you go from Shibuyu Station to Uneo Station, it costs 190 Yen. When you buy the ticket, you'll see this option displayed on the screen. You insert your money and press the 190 button. Easy peasy! Also, like when using the Underground in London, keep your ticket for entry into the subway and for exit out.
If you're going to use the subway a lot on any given day, choose an all day pass, which will set you back about $10. This can be a great deal. Keep in mind, you won't be able to use a subway pass on the JR line, but even if you have to purchase a JR ticket or two, you'll probably come out ahead.
On most trains, there are announcements in Japanese AND English. There are also either running digital displays that flash the names of the stops in Japanese pictographs PLUS English writing, OR there's a display of the line over the doors that will show you where you are by turning on a light beneath each stop as you arrive at it.
I met two New Yorkers during my travels, and we hung out for a few hours. They told me they thought the system was a little more complicated at first glance from the subways in New York City. However, we all agreed that once you got the hang of it, the Tokyo system is very robust and a great way to travel.
How about other options?
I will say I did not use the buses at all in Tokyo, but I did use them in Kyoto. I found them to be a much bigger challenge. Why? Announcements weren't in English. If there was a digital sign in the bus, it only flashed the Japanese pictographs, no English. I didn't have an English map at first, and this was a disaster. The Tokyo subway was definitely a lot easier to use than that!!! (Of course, if you go to Kyoto, just be smarter than me and get an English map from the get go.) I would assume the buses in Tokyo are similar to Kyoto. Anyway, I can't imagine you needing to catch a bus.
I never used a cab, but the aforementioned New Yorkers did. They said it was easy though---like in all big cities---a good deal more expensive than other transport. At certain times of day, you're paying for comfort. Beware, however. Cars don't always move faster than trains!!! If you DO take a taxi, the meter starts with a base rate around $7 and then ticks up according to distance traveled plus time sitting in traffic.
By far, the thing you'll do the MOST is WALK. While I also managed to get lost a few times on foot, I would always eventually run across a sign for a tourist attraction that had English writing as well as Japanese pictographs to point me back in the right direction. If I just stood there looking especially out of sorts, I found a kind Japanese person would inevitably stop and offer help in English.... normally businessmen in suits. In fact, whenever I asked anyone for help, they tried to oblige me even if there was a language barrier. Here my phrase book was handy. And I carried a pen and extra paper in my purse so that I could ask for a drawn map or a picture of the appropriate pictograph. This worked well.
I hope that I am as kind to and patient with foreigners as the people in Japan were to me.