The general duties of Pilot Flying (PF) and Pilot Monitoring (PM), also called Pilot Not Flying (PNF), are e.g. outlined in the following section of the Boeing 737 NG FCOMv1 (NP.11.2 Normal Procedures - Introduction, emphasis mine):
Preflight and postflight crew duties are divided between the captain and first
officer. Phase of flight duties are divided between the Pilot Flying (PF) and the
Pilot Monitoring (PM.)
Each crewmember is responsible for moving the controls and switches in their
area of responsibility:
- the phase of flight areas of responsibility for both normal and non-normal
procedures are shown in the Area of Responsibility illustrations in this
section. Typical panel locations are shown
- the preflight and postflight areas of responsibility are defined by the
"Preflight Procedure - Captain" and "Preflight Procedure - First Officer."
The captain may direct actions outside of the crewmember’s area of responsibility.
The general PF phase of flight responsibilities are:
- flight path and airspeed control
- airplane configuration
The general PM phase of flight responsibilities are:
- checklist reading
- tasks asked for by the PF
- monitoring taxiing, flight path, airspeed, airplane configuration and
PF and PM duties may change during a flight. For example, the captain could be
the PF during taxi but be the PM during takeoff through landing.
So monitoring instruments is primarily the responsibility of the Pilot Monitoring (hence the name). But of course the pilot flying will also monitor instruments like e.g. airspeed indicator when hand flying.
The exact duties for each phase of flight are also outlined in the FCOM, but they are of course aircraft specific. The following is part of the 737 takeoff procedure:
If you are interested in more details, I suggest you try to find an FCOM (Flight Crew Operating Manual) for the aircraft you are interested in online. It should have detailed lists of responisibilities for each phase of flight.