Delta, Virgin Atlantic, KLM, Air France and SkyTeam partner airlines have round-trip flights from Dallas (DFW/DAL) to Rome, Manchester, Copenhagen, Lisbon, Barcelona, Venice, and Madrid (FCO, MAN, CPH, LIS, BCN, VCE, MAD) for $541-$671 with a connection. Flights September to February. Most require destinations require a 7-day minimum stay. Venice availability is very limited.
A Carry-on bag and also a personal item is allowed for free. You can avoid paying for seat selection by accepting the randomly assigned seat at the gate.