Japanese Taxis work similarly to other countries. The car will have the word TAXI in English on it somewhere and they can be flagged down, or more often simply approached while parked at the side of the road. You tell the driver where you want to go and he takes you there, with a meter counting up the cost on the way. Prices are broadly in line with western countries.
One difference with western taxis is that the doors of Japanese ones are automatic. Don't try to open the door yourself, the driver will do it for you with a button in the cab. Similarly he will open it for you when you arrive and have paid.
If you have large bags with you the driver will usually be happy to put them in the boot (trunk) for you. The inside of every taxi is immaculately clean so try not to get it dirty by throwing suitcases on the back seat or letting your umbrella drip everywhere.
If you want to go somewhere you can usually just say the name of the place to the driver, perhaps appended with made (まで) meaning "to". For example, "Asakusa made" (to Asakusa). You can also point to a map and say "ここまで".
To ask how much it will cost say "ikura desu ka" (いくらですか).
When you reach your destination you can ask the driver to stop and let you out by saying "koko de ii desu" (ここでいいです).