How are pitch angle and pitch rate measured in an airplane?
The pitch of an aircraft relative to the horizon is observed on the Attitude Indicator which also displays roll angle. These are required for IFR flight but not VFR flight here in the US. As such they are found on most but not strictly all aircraft. These are typically gyroscopically based. 



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Here is a basic cutaway of the interior showing where the gyro is mounted. They are often vacuum (or positive pressure) driven but can be optical or electrical as well. 



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Angle to the relative wing (angle of attack) is shown on an angle of attack indicator. While not required these are becoming more popular. These are controlled by a sensor on the outside of the aircraft. 



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The sensor that measures this on the exterior looks like this: 



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As far as I know there is no pitch rate info on your standard panel.