The rules are different for taxis than for buses and trains.
To get a taxi is to secure the services of a taxi for transportation. To take a taxi is to ride in one. That is, you have to get a taxi before you can take a taxi:
You'll never get a taxi in this neighborhood/at this time of night/in this weather
We got a taxi at the airport and took it to the hotel.
On the other hand, it's pretty unusual to get a bus or a train (though not impossible). It's more common to take or catch a bus or train. As you can see in this ngram, the forms with "get" for buses and trains are much less common.
