With all of the train companies and rail lines mentioned in this thread, do they all congregate at one large stationSome do and some don't.
For the sake of explaining it all, you really don't have to worry about it much.
Most of your long distance trips (Tokyo-Nagoya-Osaka) will be done using the Shinkansen, all Shinkansen are operated by JR companies.
So if you want to go from Tokyo to Nagoya you go from Tokyo Station to Nagoya Station, and from Nagoya Station, the Shinkansen gets off at Shin-Osaka Station.
If you browse the Japan guide pages on Transportation for each city it is explained in those pages with all your options.
What I recommend for Nagoya - Osaka if you decide to go is purchase a discount ticket, it is known as the
Puratto Kodama Economy PlanNormally a ticket from Nagoya to Osaka would cost 5670-6300yen, however with this discount ticket it will only cost you 4300yen, the ticket has to be purchased one day in advance and the train is only 10-15 minutes slower then the fastest train between the two stations, so its certainly a deal to get:
Here is the page in Japanese:
http://www.jrtours.co.jp/kodama/nagoya.htmlThat page is the prices if departing from Nagoya Station
名古屋 Means Nagoya
新大阪 Means Shin-Osaka
You can see the schedule at the bottom of the train, I think that part is pretty much self-explainatory. Just print out that page and show it to the ticket seller at any JR Tokai Tours office in Shinagawa Station (in tokyo) or Tokyo Station, or inside Nagoya Station, remember you have to purchase it a day before.
You can also use the same plan from Tokyo-Nagoya if you wanted to save more money, however note the trip is 3 hours between Tokyo and Nagoya instead of 1hr 45minutes by Shinkansen or 1:30minutes by Nozomi. If you don't mind the extra hour and a half travel time, it can save you about 5400yen round trip.
http://www.jrtours.co.jp/kodama/tokyo.html東京 Means TOKYO
名古屋 means NAGOYA
So lets say you did all your long distance trips by discount JR tickets, you end up saving some 9400!. And the time difference isn't too bad, especially between Osaka and Nagoya. Remember the Kodama tickets have to be purchased 1 day in advance before you travel.
From Tokyo to Saitama (not sure where in Saitama you want to visit, so I'm using Omiya Station in Saitama as a fare base. There are many train lines from Tokyo to Omiya Station. The cost is 540yen. The trip takes about 40-50 minutes by regular JR trains. There are faster Shinkansen trains that do it in 25 minutes but they are more expensive. So taking regular JR trains is your best bet.
Hope that helps.