Delta has round-trip flights from Dallas (DAL/DFW) to Rome, Valencia, Venice, or Zurich (FCO, VLC, VCE, ZRH) for $546-$574 with a connection. Flights April, May, August, September, October, and November, but exact dates vary depending on the destination. No Thanksgiving dates.
Valencia is limited to only Oct-Nov dates.
A Carry-on bag and also a personal item is allowed for free on Delta. You can avoid paying for seat selection by accepting the randomly assigned seat at the gate.