To bring this all together:
You can check in for your Southwest flight at Guest Relations at any WDW resort (whether you're staying there or not), or at Guest Relations at any theme park, or at Guest Relations at Downtown Disney. This service is free to all Guests.
You can also check in for your flight using any web-enabled cell phone or mobile device. Just go to mobile.southwest.com. Even if you don't have a web option on your cellular account, your phone may be web-enabled, and you might be able to check in and pay by the minute; most Verizon customers have this option (check with your cell provider to see for sure and to confirm the procedure).
To check in for your flight, you will need all of your confirmation numbers. It's a good idea to write them on a business card and keep them in your wallet or wherever you keep your park passes, or save them in your phone or PDA.
When you check in for your flight, you do NOT have to print your boarding passes. You can print your boarding passes when you get to the airport, either at the bag check line or at free-standing Southwest kiosks. Printing your boarding passes in this manner will not change your boarding group or number; the boarding group and number is assigned when you check in, and won't change if you print or re-print your boarding passes later.