Start up torque on DC motors
With no load coupled to the motor, the torque applied to accelerate the motor's inertia is the torque of the torque-speed capability characteristic. The current will be whatever current corresponds to that curve. If that is too much current, it is necessary to limit the current and reduce the torque by reducing the voltage or putting some resistance in series with the motor.

With a load coupled to the motor, the torque applied to accelerate the motor and load inertia is the torque of the motor's torque-speed characteristic minus the torque required to drive the load at a given speed without accelerating the inertia. That is the load's torque-speed requirement characteristic.

Here is an illustration for an induction motor. The shape of the curve for a DC motor is different, but the principle is the same.

Added detail:
The motor produces the torque given by its characteristic curve whenever it is not operating at the intersection of the motor curve and the load curve. The difference between the torque given by the motor curve and the torque given by the load curve is applied to accelerate (or decelerate) the load to the intersection of the two curves. That is the normal operating point. The diagram has been revised to show that more clearly (I hope).