^^ IAC Valve would be worth a look.
With engine idling unplug the IAC valve's connector, idle should drop to approx. 500, then plug IAC back in, if engine is warm idle should go to 750, if cold 1,000.
If this doesn't happen remove IAC and clean it.
If heater is on Defrost and you have A/C then the A/C will be on which can kick up the RPMs a bit.
Put heater to floor or something else and see if changing RPMs go away, if so then could be an issue in the A/C circuit.
Computer for manual trans is setup to hold the RPMs high when shifting gears, also RPMs shouldn't drop below 1,000 until speedometer gets below 5mph, these are fuel saving and emissions settings.
As far as climbing RPMs I would check the throttle linkage specifically its spring return and the cruise control cable if so equipped