When you move to a new city, it takes time to explore the area, find all the interesting places, and get settled in. Each form of transportation has its pluses and minuses, and if one form is not available, there'll be whole sections of the city that you can't get to.
Having a car is nice, but for sightseeing, the primary drawback is that you're not likely to simply stop it, get out and walk around. Scenery whips by too quickly, and you become too focused on just getting from point A to point B without paying attention to anything in between. Naturally, living in Tokyo, there's also the additional problems of little parking, narrow winding streets, traffic jams, and $5/gallon gas prices. Doesn't matter much, I don't have a car and no plans to buy one. However, the real plus is that you can choose to get out of the city at any time (like, to Mount Fuji), and visit places that aren't near the train stations, while carrying lots of luggage and food.
If a target destination isn't near a station, then the fallback plan is to take the bus. Tokyo has a very good bus system, but it is confusing. Plus, if you're going more than 5 miles, the costs start to add up. There's also the disadvantage of not being able to get the driver to take you directly to your destination, if it's not on his route. But, if you have access to trains and buses, you don't need a car. Again, though, you're focused on destinations, and not on exploring the areas in between.
Taxis - If you're on a fixed budget, taxis are not a viable option. But, they can get you directly to your destination, like buses can't.
Trains are great. You can get almost anywhere in Japan by train. It may take a few transfers along the way, but you can get there. The problem is that the costs add up fast. From my apartment, it's $5.60 one-way to get to Akihabara. And, it's $1.30 just to go from one station to the next, even though they may only be 1/2 a mile apart. So, doing a lot of sightseeing along any given train line does get expensive. This is where the monthly pass comes in. The second you get a job, your employer pays for transportation costs. Buy a monthly pass, and not only don't you have to buy tickets every day, but every stop on that line between your home and your office becomes free. If you live far enough away from your office, you can now do a lot of exploring. This is a very good thing.
Walking is kind of obvious. Go somewhere, walk around, go back. If you can get to that "somewhere" by train or bus, you can explore as much as you like. You can now wander around aimlessly, stop and take pictures, stop in a shop just to see what they have, etc. With walking, you can really get a feel for a neighborhood like no other form will let you. The problem is that you may only have a 2-mile exploration radius, and you're not going to go up and down every single street simply because it takes too long and you'll get too exhausted. But, couple walking with a train pass, and you can see a lot of things for free. Without the train pass, you're pretty much stuck with a 2 to 4-mile radius circle around your apartment.
Which brings us to my main point. Buy a bicycle and your exploration range explodes. You can't really bring a bike on the train, but some buses do have carrier racks on the front. Even if you just stay around your home base, you can go about anywhere within a 25 mile radius as part of a day trip. (Tokyo Station is 20 miles from my home; Kawasaki city is 15 miles). As long as you can survive the twisty little roads, the need to ride on the sidewalk most of the time, and the cars and the pollution, you can get almost anywhere for free. The main drawback is having to worry about someone stealing or damaging the bike. But, you'd have the same worry if you had a car.
I just bought a new bike last week. Pruning, pruning...