"going up and down simultaneously"?? two needles, lol
If temp gauge goes up at idle(stop light) and then goes down when moving again then fan clutch is probably bad.
All fuel injected engines use an IAC(idle air control) valve to set idle.
This is an electric motor on the upper intake controlled by the computer.
It opens and closes an air valve that by-passes the throttle plate.
These can get dirty and stick open raising RPMs, but they can be cleaned.
I would warm up the engine first, then unplug the IAC Valve's connector, idle should drop to 500 or engine may even stall either is good because it means it isn't stuck and you don't have a vacuum leak.
If idle stays high then IAC could be stuck OR you could have a vacuum leak.
PCV Valve hose is a common hose to crack and leak, check it carefully.
Also with engine warmed up and shut off, try spinning the fan, it shouldn't spin, you should be able to move it but not spin it, if it spins fan clutch is bad.