Instrument(s) for measuring (and recording) temperature, voltage and current
You want a digital multimeter (DMM) with a thermocouple setting. And a thermocouple sensor that works with it. 

If you're spending enough to get the thermocouple function, you can likely get a DMM that does its own recording at no additional charge, so no Arduino or other device would be needed to achieve that function.

If you want to avoid spending a few hundred dollars on a known-accurate DMM, you can investigate making your own cold junction circuit to allow reading the thermocouple with any voltage meter (or the ADC function that's probably included in your Arduino) you like.