Auto Insurance Claims: Switching auto coverage to a new vehicle, full coverage insurance, auto coverage

Question
I just purchased a newer vehicle, but want to shop around for the best price on full coverage insurance.  The car salesman said my current insurance will cover the new vehicle for thirty days, he called it a grace period, is he right?  Restating it, if I would have an accident, heaven forbid, would my old policy cover my newer vehicle?  
  Also, do you carry uninsured and underinsured motorist on your policy?  
Thanks

Answer
Most policies do allow for coverage on a newly acquired vehicle for a certain period of time.  But not every policy is the same so I would not know without reading your policy.  Your agent can answer this question.

I do carry UM and UIM - they are very important to have.