I'm really new and inexperienced with electronics. For my work, I have to use an SBG01 heat flux meter (https://www.hukseflux.com/uploads/product-documents/SBG01_manual_v2023_0.pdf). It produces a voltage in the range of 0-40 mV. I own an Arduino Uno and I wanted to use it to measure the voltage and log it to my PC. However, the precision of the input ports on it is not sufficient enough.
What would be the simplest way to do this with the precision of at least 0.1 mV?
As I mentioned, I'm really new to electronics and the simplest plug-and-play option would be the best.
Not possible with just an Arduino. You need a 15 to 16 bit ADC module, such as the ADS1115, and pay very careful attention to circuit construction techniques (shielded wires, enclosures and proper grounding).
Even simpler: buy the readout unit designed for that sensor.
One option in the Arduino UNO/Nano is the INTERNAL voltage reference. It's a stable 1.1V plus or minus 9%. When your sketch is running you can measure the actual value on the "AREF" pin. That will give you a step size of about 1.074 mV.
Other reference values are available on other models:
If you REALLY need 1mV steps, get a 1.024V precision voltage source and connect that to AREF as an EXTERNAL reference.
And before you ask: no you can't put a 0.512V reference on and get 0.5 mV steps. The A/D only works with references down to 1.0V. That's probably why they designed the average INTERNAL reference to be 1.1 instead of 1.0. If the average was 1.0 and the limit was 1.0 they would have to throw out about half of the parts because they landed below the limit.
You can built an amplifier with an op-amp and two resistors BUT things get tricky with low levels because of the signal-to-noise ratio. Any existing noise gets amplified, the circuit can pick-up stray electromagnetic noise, noise can get-in through the power supply, and the op-amp itself generates some noise.
Op-amps also have SOME DC offset which you might have to "deal with" if you need to read down to around 1mV.
You'll also need a "rail-to-rail" op-amp that can go all the way to zero, or it's more common to use + & - power supplies so the output can go to zero (or negative).
It's NOT IMPOSSIBLE.... Microphones put-out a few millivolts and a mic preamp sometimes a gain of 1000 but a high-quality mic preamp is not an easy DIY build.