First off, my disclaimer: I was not there so I will comment based on speculation only to ease your concern.

Sometimes, due to tire wear, the nose wheel assembly can become slightly unbalanced. Also, because of the altitude of the SLC airport, the airplane may have to be accelerated to a higher ground speed for takeoff than other lower altitude airports. Those two things combined with the small diameter of a nosewheel at 140-150 knots (estimates) can create a loud vibration originating at the nose gear once it is lifted off the ground.

You also mentioned that you were in line behind other airplanes waiting for takeoff. From what you described, this sounds like the classic wake turbulence encounter. Again, my disclaimer: I wasn't there.