I book all our long haul flights to the USA and my daughter’s as she flies back and forwards to Europe. I understand your concerns completely as I was just the same. When you have completed the booking, you are given an E ticket number. This number is in effect your ticket. Print a copy of the receipt out and just take this along to the check in area, or self serve kiosk ( your preference) and your boarding passes and the return tickets are printed for you there and then.
You will find besides the e ticket receipt in the SWA web site, they will send a confirmation email as well. If it makes you feel more secure print that too, although it will only have the same info as the receipt.
With Virgin, once you have the e-ticket you can type the number in and make changes, like where you’re sitting, or what meals you would like etc? ( might be different with SWA)
Dont be nervous, its really very easy, take your time and double check, dates times etc and then you’re all set-