Usually I look up tickets on a wholesaler's website (Travelocity, Expedia or Orbitz) and then I go to the website of the airline who is offering the cheapest fare on the wholesaler's website. Sometimes it's still a few dollars cheaper, but not always.

Most airlines used to offer extra miles for flights booked via their websites, but I think none of them are still doing it. If you have an airline credit card, you are very likely to get double miles when purchasing from the airline's website, although it's not the rule. I never buy tickets over the phone because they always charge a handling fee or via travel agents, except wholesalers.