Will a driver stay at your location until you are done? Will they wait for me?

You should ask your driver whether it is ok to make stops for you to run errands or to pick up additional passengers.  The duration charges will be incurred and they may be ok with waiting for you.  

Uber also offers an ability to specify multiple destinations for your trip.  With this, you can enter up to 3 stop locations along the way to the final destination.  This feature it meant to be used to run small errands and pick up additional passengers.  Uber however, strongly suggests that the riders keep these stops very short, say only 3 minutes at each stop.  This is to be courteous to the drivers.