Can I book a ride and make a quick stop on the way?

Yes, you can request the driver.

Yes, the simplest way is to request the driver to make additional stops.  The other way is to use a feature within the app (assuming you are on Lyft or Uber), and you can specify multiple destinations.

There are no extra "fees" to make stops; however, you will have to pay for the additional time and distance that results from making changes to your trip.  You should also be aware that you may forfeit the upfront pricing, i.e. "quoted price", that you received at the beginning of the trip.

How long you stop and make the driver wait should be discussed with your driver.  While the driver does get paid for the time waiting, on a busy night, it is not an efficient way for them to make money.  You should be considerate of this if you ask the driver to wait, and if you feel it'll take a long time, you could end the trip temporarily and request another Uber.

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Uber Driver, CEO of TaxiFareFinder and Unleashed, LLC