I would agree with Hartington. You will also get somewhat better service on BA or Virgin or Air France or Lufthansa (well, not always with them) than you will from any of the US carriers. DO NOT use a BA flight that is a code share with American Airlines, though.
My suggestion would be BA to Phoenix, as immigration there is usually pretty quick, unlike Chicago, New York, Miami, etc. Then drive or take South West or failing them, US AIR to LAS