Fares for regular train, bus, and subway systems are not expensive. However, tickets for the Shinkansen (bullet train), and express trains can be very expensive.
The only express train you might need to take is the Narita express, which takes you to and from Tokyo station and Narita airport.
I don't recommend the Narita express, it's not convenient if you are traveling with heavy luggage, and, depending on your arrival time, a seat might be hard to get. You should take the airport limousine bus instead, it is as fast (or faster) than the train, as well as being more comfortable and less expensive.
The ticket counter for the bus is to your left after you clear immigration and customs. The cost for a ticket is 3000 yen, and there are porters who will take care of your luggage for you.
Take the bus to Tokyo Station, it will drop you off outside the station, just across the street. You'll find a taxi stand in front of the station, you should take a taxi to your hotel from there. Meguro is not that far from Tokyo station, so the taxi fare will not be expensive.
Once you are set up in your hotel, then you can venture out onto the train system. Meguro station is on the Yamanote line, which will take you to most of the places you have on your list. It runs in a circle around the city, with stops at Shibuya, Harajuku, Ikebukuro, Ueno, Akihabara, and Tokyo Station.
To use the train, first find the ticket machines. Above the ticket machines is usually a large map, showing the various stations where the train stops. Next to each station on the map will be the train (or subway) fare in yen. Put that amount into the machine, and press the single fare button (which is pictogram of a single person), and the machine will give you a ticket. Put the ticket in the slot at the entrance to the train gate, and pick it up on the other end after you pass through.
If you don't have exact change, insert a bill into the ticket machine, the display will then show various fare amounts. Touch the screen on the fare which matches your destination, the machine will give you your ticket and change.
Here is a map of the JR train system for greater Tokyo
http://japanrail.com/pdf/railpass/greater_tokyo.pdf This map is similar to the ones you'll see in the train stations, except that they will have the fare amount listed next to each station name.
To get to Disney, you take the Yamanote line to Tokyo Station, and then follow the signs in the station to the Keio line. It's a long walk (Tokyo Station is huge), but all signs are bilingual in English/Japanese, so getting around is fairly easy. You can take the Keio line to Maihama station, which is where Disney is located.
Don't bother with the 100 yen shops, they sell mostly household/kitchen items. If you want souvenirs you should go to Asakusa, which is an old part of Tokyo, and home to the Sensoji Temple (a must see). To get to Asakusa, take the Yamanote train to Shimbashi, then you can transfer to the Asakusa Subway. Get off at Asakusa station, the station exits near the entrance of the Sensoji Temple, and the thousands of shops which surround it.
You don't need to buy a one-week train pass for your stay. As I said at the beginning of my reply, train, bus, and subway fares are cheap. You'll probably spend less than 1000 yen in train fare on your busiest day.
Have fun during your stay, you can pm me if you have any questions.