Is there a grace period after you get your licences for getting insurance?

You usually need insurance before you purchase or lease a vehicle (especially if you buy it on a loan).
You can have a licence and never get any car insurance if you never purchase or lease a vehicle (if you only rent vehicles always accept the collision damage waiver, but this is not liability insurance).
You never have to get car insurance if you post a "liability bond" with your state's motor vehicle department (but this bond can be a half million or more depending on the state you live in).
You can also "self insure" and accept all liability and risk of lawsuit in the case of accident.
No, you do not ever have to get car insurance... but you will be a fool if you don't.
Personally, I got my driver's licence when I was 14 years old and did not get car insurance until I was 21 (my parent's insurance covered me in High School and College) when I was applying for a loan to purchase my own car 6 months after I started working. The lender would not approve the loan until I showed them documentation that I had purchased insurance that met their minimum requirements (which were higher than the state minimums).
As far as the licence plates are concerned, most states will not let you register the car without proof that you have either valid insurance, a "liability bond" on file, or proof that you are financially capable of "self insuring". You will not be issued licence plates until the car is registered.