Just like in other major cities, getting to know the public bus system can be quite a hard task. The Tokyo Metro Government runs most of the buses within inner Tokyo ("Toei Bus") while a number of private operators run services in the suburbs.
For nearly all locations within the central 23 wards of Tokyo, you'll have no problems doing a trip with the use of the JR or Metro trains only, but for those who really want to travel like a local, here are some tips:
How to use them
Board from the front. Put your fare (a flat fare of 210 yen) in the box next to the driver. If you have a 1,000 yen bill, there is a slot where you can insert your bill and it will automatically give back your change. There is no ticket or receipt, simply proceed further and find a seat.
(Note that Toei is the only operator that has the auto-change function. Most other operators have just a money changer that will break down your bill to smaller coins. Also any bills over 1,000 yen can't be used so be careful!!!)
If you have a PASMO or SUICA card, you can use them also; just tap on the sensor located next to the driver when you board. You don't need to touch it a second time when getting off.
When you want to get off, push a button near you and exit from the door in the middle (rear) of the bus.
For travel outside downtown Tokyo with different operators, different rules may apply.
Suburban Tokyo (outside 23 wards/Non-Toei operated bus): Usually boarding from back and exiting from front. Distance-based fares, pay as you exit. For cash, take a small paper ticket upon entry (It's proof of where you got on). For IC card, tap both when getting on and off.
A limited amount of services do have a flat fare structure and in that case you would board and pay at the front door.
Yokohama: Board from front, exit from back. Flat fare of 220 JPY to be paid on boarding. For IC card, tap once upon entry.
Kamakura: Most services board from back, exit from front. Fare is distance-based, pay as you exit. For cash, take a small paper ticket upon entry. For IC card, tap twice both when getting on and off.
Hakone: The rear door is not used. The front door for both getting on and off. Fare is distance-based, pay as you exit. Tap twice for IC card. For the Hakone Free Pass, show it to the driver when getting on and off.
Nikko: Most buses board from back and exit from front. Some longer distance buses have only one door at the front.
No flat fare, so in any case pay as you exit. For IC card, tap the card twice upon entry and exiting
For the systems with the distance-based fares, normally you have to take a small ticket that has a number printed. There is a monitor in the front that will show the fare corresponding to these numbers. The IC card such as PASMO or SUICA eliminates the need to check fares.
Which routes to use
Now here's the tricky part - knowing which routes to use. What's confusing for non-Japanese readers is that the routes have a prefix in Chinese character (kanji) followed by a two-digit number. For instance, at Shimbashi station there is a 都01 going to Shibuya Station and a 市01 headed for Toyosu Market! There is small English signage of the destination at the front of the bus, so you'll have to examine that carefully.
Google Maps will show various bus options where subway/trains are not available. It is probably easier than trying to navigate with a printed map. Usually the timetable is accurate although occasionally unexpected route diversions take place which is not accurately reflected.
Of all the routes out there, 都01 running between the Shibuya Station east bus terminal and Shimbashi Station stops by the Roppongi Hills which is convenient since you would have to change at least once if using the subway. The RH01 runs a direct route between Shibuya and Roppongi without making stops, so it is the fastest way to travel between the two locations.
The 都08 originates at Kinshicho Station's north terminal (connection to JR and Metro Hanzomon Line)and runs along the boulevard south of the Tokyo Skytree, with stops at the Asakusa Station before terminating in Nippori Station which is a good base for exploring the neighborhood of Yanaka.
The 海01 routes starts at Monzen Nakacho Station of the Metro Tozai Line. It makes a stop at Toyosu Station (Metro Yurakucho Line) before ending up in the Odaiba District of the waterfront. It's a cheaper alternative to the expensive Yurikamome Line, though the views will be far less impressive and would take double the amount of time.
All of the above routes have a frequency of one service about every 7 or 8 minutes; other routes might have less with only 2-3 services an hour. The buses stop running far earlier than the trains, with the last one around 10pm.
The Tokyo One Day Ticket (1,590 yen) or the Toei One-Day Economy Pass (700 yen) will work on the Toei Buses, but the Tokyo Metro Pass will not.
There are some additional info available on the Toei English site. http://www.kotsu.metro.tokyo.jp/bus/routes/eng/
The Tokyo BRT (Bus Rapid Transport) service is operated by a separate company but has the similar rules as Toei with a flat fare (220 JPY) and rule of boarding from front.
Originally planned as a way to transport personnel for the Tokyo Olympics, the buses connect Shimbashi station with the Tokyo Big Sight convention center.
They are a cheaper (and in most cases also the faster) alternative to visiting the Toyosu Market, Ariake Arena or the convention center rather than using the Yurikamome train that does the similar route.
Frequency is about every 10 minutes. Half of the services only go as far as Toyosu. About one service an hour uses the 18 meter-long articulated bus which is an interesting experience.