In Japan, there are many types of transportation you named it: shinkansen, subway/metro, bus, taxi, etc. The most popular one for tourists is JR Pass; you can use it to travel through all cities in Japan. They have 7 or 14 days pass and should be used in consecutive days. This pass can be booked online through
JR Pass website or official travel agents in your country. BUT you should pay attention that they also have regional passes! I often browse the website but didnt realise it until a week prior to my departure. We went to Japan for two weeks and decided to only buy a 7-day JR Pass. You can reserve seat too with this pass by coming to JR Office in the stations and search for the schedule you want. Before riding the shinkansen, it's best to take a look which car is for non reserved and reserved seats they put in on board because if you ride the wrong car, you should pay the fine.
If you decide to go to some cities which are close to each other, JR Pass maybe isnt your best choice. For example, I went to Miyajima, Hiroshima, Okayama, Osaka in the first five days then after reading here and there, JR Kansai Hiroshima Area Pass would be my best choice because I needed JR Ferry to go to Miyajima and with this pass, I can ride either Nozomi or Mizuho shinkansen. These shinkansens are the fastest as they stop in mostly main stations compare to Hikari or Kodama. They arent covered by the ordinary JR Pass; by using JR Kansai Hiroshima Area Pass I can save quite some time. Hikari runs in the similar track with Nozomi but stops more. To see which type of pass suits your trip the best, you can click here.
Besides JR Pass, there are also other railways run by private companies such as Kintetsu, Hankyu, etc. In my Nara day trip, we used 1-day Kintetsu Rail Pass as Kintetsu station was closer to Nara city centre and you can use it in Osaka, Nara, and Kyoto. Japan really give tourists lots of benefits; there are many local passes for one day, two day, three day, or five day provides with discounts to tourist spots if you show them your pass before your payment. You can read more about it when I explain it through the day trips.
JR Pass doesnt always cover the city area so using IC cards will be the best choice. This card can be used in subways or buses and I really recommend it since it's also rechargeable. Sometimes it takes time if you always pay cash yet at the same time, you only have big amount of money. Just tap it at the station entrance and tap again at the exit. It's also accepted in paying for coin lockers in station, vending machines, shops, and restaurants. Most popular one will be ICOCA, Suica, and Pasmo. ICOCA can be bought in Kansai area, whereas Suica and Pasmo can be purchased in Tokyo. Now these cards can be used in whole Japan so it's really convenient. One thing I learnt during my trip was it's the best to get the city map when you arrived in the main station so that you could choose the most efficient and budget friendly transportation.
Note: Everything I share here is based on my own experience. Maybe it's suitable for me but not for you so please read and search for more information. Let's enjoy the trip!