Sounds like a standard type K thermocouple (there are also probes specifically designed for fluids) should work fine, and is supported by your DMM. If you want better accuracy, you can use other types of thermocouple or resistive probes, but these aren't supported by your DMM, so you need a separate thermometer (or one with integrated probe).
Not sure how well IR would work, you have to correct for emissivity, the specs usually state the emissivity it was calibrated for. Liquids would probably screw that up, you may have to measure it indirectly (measure something solid in contact with the liquid), which will increase the heat capacity of your 'probe'.
When measuring liquids, keep in mind that you either need to stir, keep it in an enclosed space with the same temperature for a while, or accept temperature gradients. Same with solids actually, except that they are usually harder to stir.