In Colorado I have 60 days to register the car,?

bought from private party. Exactly how long am I able to drive it without the insurance if I just bought it a day ago ?

Other answer:

You have zero days to drive without insurance except on private property with the property owners permission.

You should register as soon as you can, however penalties don't start for failure to register until 60 days have passed. From the website:

Registration Required

Within sixty days after purchase (42-3-103(1)(a) C.R.S.)
Within ninety days after becoming a resident of Colorado
An owner of a foreign vehicle operated within this state (42-3-103(2) C.R.S.)
Every nonresident person who operates a business within this state and owns and operates in such business any motor vehicle trailer, semi-trailer, or trailer coach (42-3-103(3) C.R.S.)
Within forty-five days after the owner has returned to the United States (42-3-103(4)(b) C.R.S.)

Zero days. It's illegal to drive it at any time without insurance. You should have arranged insurance before you bought it.
In America, You can NEVER drive a car without insurance not one day, not one mile.
J E T H R O:
You have zero days to drive without insurance. Don't move it until insured.
You better park it till you get insurance. Your breaking the law and taking a serious risk.
I would not have it on the street without insurance, that is needed from the start.
Scott H:
Legally it can't be on the road at all without insurance. There is no "grace period".