If you travel frequently, you no doubt collect FF miles, so it is best to book on the airline's website. On Delta for example, you get 1000 bonus miles for booking on the website, and if you use the Delta AmEx card you get double the dollar cost of the ticket in miles. (Plus you get actual mileage for the flight, plus elite bonus if you are silver, gold or platinum)
However, if the price appears too high, I visit expedia.com to find me a cheaper DL flight, just to satisfy my employer. (DL is usually not the lowest cost, but is generally within $25, unless the flight is fairly well booked up.) Then I go back to the DL website and book the flight that expedia found.
For rental cars, I use the Hertz website and get cars even when my corporate travel agent says they are all booked (I have a Hertz gold card). For hotels, I belong to the Starwood, Marriott and Holiday Inn clubs, so I book one of those on their websites. Sounds like an ordeal, but I do all that at home in my spare time in less than 30 minutes. Since I have a son in college, we can use all the miles I can earn.
"In God we trust, everyone else bring data"