Options to get to the Swan:
1) Uber or Lyft. These are referred to as "rideshare." You need to know that term because when you use them, the sign for the pick-up spot may say "rideshare." (Not sure what the sign says at the airport.)
2) Taxi.
3) Shuttle. This will stop at hotels other than just yours. Mears is the company name you will hear the most. Go to MearsTransportation.com for information.
4) Towncar.
FL Tours (a sponsor of this board), Tiffany Towncar and Tony Hinds are the names you will see the most. They pick you up at the airport, help you with your luggage, and take you to your hotel. They will stop at the grocery store on the way if you arrange it in advance.
5) Minnie Van. The drivers are Disney employees. I believe you need to call to arrange airport transportation. If you are not going to/from the airport, you book a ride using the Lyft app. Two car seats are included, which is a reason some people use this service.
6) Rent a car at the airport. That would allow you to go the the grocery store,
Walmart, or anywhere else you want. If you don't want to keep the car for your entire trip, you can drop your people and bags at the Swan, then return the car at the Dolphin -- but I'm not sure if that is only for certain brands, so investigate first. You probably know that the Dolphin is right near the Swan.