Contactless debit/credit cards are accepted for payment on the Osaka metro, but not in Tokyo or Kyoto so you will need to get an IC card (Suica / Passmo / etc). It does not matter which one as they are all accepted interchangeably.
- If you have an iPhone, you can get a virtual Passmo card in Apple Wallet and recharge it via Apple Pay.
- Otherwise, you will have to get a physical card. (Google Pay technically supports IC cards, but only if your phone was bought in Japan so you won't be able to use that). You can pick up a Passmo Passport at the JR ticket office in Haneda airport. Warning that the tourist IC cards (PASSMO Passport / Welcome Suica) are only valid for 30 days and there is no way to refund money loaded on to them, so towards the end of your trip make sure to try to spend all of it and not load on too much. You probably won't be able to get a normal IC card, just a tourist one. You can load cash (Yen) on the card at train ticket machines as well as most convenience store ATMs. To get cash, I recommend just using your foreign debit card at a convenience store ATM, although make sure you have a card with low/no ATM and foreign exchange fees.
Once you have an IC card, you can use money stored on it to pay for just about any train in Japan that is not a Limited Express -- for the limited express you also need a separate paid reservation for the specific train (and will likely want a seat reservation). It can also be used to pay for most busses, vending machines, and to pay at most places that accept credit/debit cards.
For Shinkansen, I think you can use your IC card to pay so long as you ride in an unreserved seat. I haven't actually tried this and am seeing conflicting information online; you might need to link a credit card to it somehow? I recommend booking a seat reservation with SmartEX. You can link your ticket to your IC card so you just tap in/out. There are some discounts available if you book sufficiently in advance. You can also book Shinkansen tickets at JR ticket offices or Shinkansen ticket machines (with a blue train symbol).
For Limited Express tickets (not sure if you will end up taking a limited express train, but just in case), how to book the ticket depends on what company runs the train. There is usually a way to buy online, although you might have to go to a ticket machine to print it out. Generally, they will not check your ticket so long as you are sitting in the right seat, but you should still have it with you. You can also use a ticket machine or ticket office, although it probably has to be a ticket machine / office for the company that runs the train. The only benefit to booking in advance is in case the train might fill up, or if you don't want to deal with getting a ticket right before the train; pretty much always the price is fixed and tickets are changable/refundable until the train leaves. Generally, the ticket will be for the seat reservation and limited express surcharge, and you still have to tap your IC card on entry/exit to pay for the base fare. You can also get a ticket to cover the base fare (or one ticket that includes both), but using IC is sometimes cheaper and never more expensive, and AFAIK all train stations in those areas take IC.
It used to make sense to get a Japan Rail Pass most of the time, but they nearly doubled the prices a couple years ago and now it almost never does. The JR pass is 50,000¥ per person for one week; you will probably end up spending closer to 20,000 per person by my best guess.
