Difference between "take a taxi" and "get a taxi"

Solution 1:

In the realm of travel, to take something is to use it as your mode of transportation:

take (transitive) to use a particular type of transport

take a bus/train/plane/taxi etc: I usually take the bus to work.

In British English, get is more common than take for this usage:

get (transitive) (never passive) to use a particular vehicle to travel somewhere

He usually tries to get the 9.03 train.

In American English, to get a taxi, etc. is to obtain its use. I might ask my hotel concierge to Please get me a car to South Station, or inquire of my host Is it easier to get a cab on Lexington or Park Avenue? Similarly, to get a flight is to reserve a seat on a flight, whereas to actually fly is to take a flight.

There are a variety of other ways to express the same— I can catch my transportation (e.g. catch the Shinjuku Line, catch a flight, catch bus #42), ride in/on it (e.g. we rode the tram back to Palm Springs), or simply go on/in/by something (e.g. Can I reach LAX by Metrorail?) among others. But not all of these may apply to all modes; we fly rather than ride aircraft, for example, even though we are passengers and not pilots most of the time (e.g. We always fly Air New Zealand to Fiji).

Solution 2:

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.