Once you download the app, you will need to make an account and add the credit card you'd like to use.
Booking a ride is pretty intuitive if you've used Google Maps or Apple maps for directions. If you're booking to and from the airport, the app will ask you where in the airport you'd like to be picked up or dropped off. You can schedule rides in advance or ask for as soon as possible. Once you enter the start and end locations, the app will quote different options, and you can pick the one you want. UberX is a normal 4 door car, XL is a larger crossover or SUV, and Black will get you a nicer car (like a Mercedes or Cadillac). Uber Share is a carpooling option that charges by the seat, and I wouldn't recommend that for an airport transfer, even if it looks cheaper.
They won't do a grocery stop. You can schedule multiple stops, but those are meant for quick drop-offs and pickups. You can have them drop you off at the store and one person watches the luggage while the other shops, or go right to Disney first and make a separate trip to the store.
Cost will be anywhere from $30 to $50, plus tip. You can rate your driver and add a tip in the app after the ride is over. I personally use Uber or Lyft every time I visit Disney World. Having my own ride on my schedule is worth the upcharge from Mears, especially when going from WDW to MCO.