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The takeoff speed of such aircraft varies quite a bit, depending on the takeoff weight and the use of high-lift devices like flaps (2) and slats. However, a good average speed range is about 160 mph (260 km/h) to 180 mph (290 km/h). Some typical takeoff speeds for a variety of airliners are provided below.
As indicated in the question, the speed of sound at 35,000 ft is 663 mph (1,067 km/h) according to the standard atmosphere model. Even if the plane's ground speed is over 700 mph, however, its airspeed will still be only 560 mph. As a result, the aircraft's Mach number turns out to be about 0.84, or 84% of the speed of sound. In other words, the Boeing 777 never flew faster than the speed of sound during your flight.