Trains.
There is the Shinkansen (high speed bullet train) between most major cities. There are also Limited Express trains between less popular destinations. You don’t need to prebook for this and can generally just head to the station 30ish mins before you want to leave and you can buy a ticket.
For everything else (subways and local trains), you can use an IC card. If you have an iPhone, you can load a Suica card through Apple Wallet. You just can’t use an American Visa Card to load this. If you want a physical card instead, they are easy to grab at the popular stations.
The trains run from 5am to midnight. After midnight is usually the only reason you would have a take a taxi as they are going to be a lot more expensive than subways or buses. There are local taxi apps you can download. If you don’t want to download one of those, you can use Uber to call taxis, it’ll just be a bit more expensive.
Really depends on how much time you have and how fast paced you want to move around. Generally I would recommend this for each city:
Tokyo: 3-6 days
Osaka: 2-3 days
Kyoto: 2-3 days
Day Trips from Osaka/Kyoto: As much as you want (maybe 3 days)
Kanazawa/Takayama Region: 5 days
If you want to do the standard trip for your first time, something like a week in the Kansai area (Osaka/Kyoto) and then a week in Tokyo is pretty reasonable. Can shorten each area by a few days to fit in a few days in another region if you want.
Cities (click each for more details)