When you buy a new car, you're supposed to:
- drive at a variety of different speeds
- avoid hard acceleration
- avoid long drives at the same speed (such as on the highway)
In American English, this is called the break-in period for the car, or more specifically for the car engine. This term is commonly used for machinery. The break-in period is the time in which you are breaking in the engine.
The term is also used for clothing, particularly leather clothing which can be very stiff until it is worn for a long time.
Note that the term is meant to evoke a physical process acting on a physical object. It does not specifically imply that the user needs to pay greater attention or act with more care during the break-in period.
As per the comments: In British English, this is called the run-in period for the car, the time in which you are running in the engine. The phrase "breaking in" is reserved for animals (such as horses).