Just recently my 2000 Ford Taurus starting shaking and vibrating while moving.

Could be a broken belt in a tire. Does it do it all the time while moving? More noise going around a right or left turn?

does it make the steering wheel wobble starting at or below 15MPH, without any odd noises? does the wobbling start to lessen above 45-50 MPH ?That would seem like a tire issue as stated in the other answer.Otherwise please give a more descriptive example of what it's doing so we can help you troubleshoot further.