Taking a taxi in Japan is definitely not as cheap as taking public transportation.
However, I must say that Japanese taxies provide very nice customer experience.

【Where you can get a taxi】

You can get a taxi most easily in front of any train station.
Going out from the station and you will see taxies lining up there.
Remember that you will have to get to the designated area for taking a taxi.

You can also make reservations by phone or app.
The taxi companies I am using most in Nagoya city are like below.

Meitetsu Taxi
052-331-2211/22

Tsubame Taxi
052-203-1212

MK Taxi
052-912-5489

All of them are available for reservation like if you are planning to go to the airport in the morning or designated time.
The taxies companies are also available with English drivers actually, but you will need to make a reservation in advance and of course, the additional fee will be charged.

If you are going to the airport, most of the taxi companies provide plans with a fixed price, which means that the fare will not change even the distance is different. However as the plans are different depends on companies, better make inquiry in advance.

【What is Japanese taxi like】

The first time when I took a taxi in Japan many years ago, I was astonished that the taxi door opens and closes by themselves!!!! :-O 
You know in my country basically the customers need to do by ourselves.
Also, the taxies in Japan are very clean.
I can remember that when I was in my country, I once took a taxi that the trunk was full of garbage and wast appliances e.g. electric fan (whyyy??)
But in Japan, it never happens.

【How can I pay】

For taxies from big companies, paying by both cash and credit card are acceptable (basically).
However when you go somewhere kind of far from the city, bring cash with you will be a better choice.

【What we can do for you】

Actually, atinn, our company, provide pick-up service from the airport.
We will make reservations for taxies for our customers, and meet you in front of the exit and go to the destination asked in advance!
Especially for someone who comes to Japan for the very first time, it is always happy to have someone you know to meet up at the airport :))))

If you are not sure how to make reservation for taxies, or if you are interested in our service,

Hope the above information helps!

Please note that the content is based on information in Dec 2017. Please contact us or check the website of the companies mentioned above for latest information.