how does one acquire a taxi cab business?

You start with your own cab and build up business by word of mouth. Once you have enough customers, buy a second vehicle and hire a second driver. With more business, you buy a third and so on.

Either that or purchase an established business.

Acquire enough money to buy it out.

This is actually another question. But can you use your own cab to work at a already established dispatching station ????? Email me or reply please thanks.