What is the standard procedure in case of inconsistent sensor readings?
The procedure used would depend on which information is conflicting/unreliable (and of course the aircraft type).

With one or more unreliable airspeed indications (which is one of the more critical ones), the initial action is NOT to start troubleshooting/figuring out which indicator is "the correct one", but to ensure safe speed/flight path by other means. The initial actions on Boeing aircraft is to disconnect automatics (auto throttle, autopilot, flight director) and control the aircraft given memorized combinations of pitch and thrust. With the aircraft under control, and when directed by the checklist, you start analyzing the situation, to see if you can rectify/isolate the faulty source. If a reliable source can not be determined, the aircraft continues to be flown by pitch/thrust settings given in the manual (based on altitude, if known, weight, flaps/gear, desired vertical profile etc).

In the case of unreliable altitude, in case you can't determine a reliable source, you can typically use radio altitude below 2'500 ft. A big caveat is that if the transponder is set to use an unreliable static pressure source, the altitude that ATC sees on secondary radar will also be unreliable. An example of this was seen in the Aeroperú 603 accident, where the aircraft took off, with the static ports taped over, and ATC was asked to assist with altitude and speed information: CVR transcript - Wikipedia

Erroneous airspeed/altitude indications often imply each other, as an unreliable static pressure source will affect both.