The output impedance is set to "High Z".
No, it isn't ... the expected load impedance is set to High Z. The generator uses this to estimate what voltage the load will see (ie. it assumes it will see the full output voltage because the fixed 50 Ohm output resistance of the generator will take nearly no voltage). You should only use this at low frequencies or when using short cables. In this case (at 60 Hz with 1K input impedance for your ADC) it's fine.
The scope has an imput impedance setting of 1Mohm (you can choose 50ohm or 1Mohm).
This is fine.
Up until now, I have just been connecting the output of the function generator directly to the ADC pin. Am I doing this wrong?
No, in this case it's fine ... the input impedance of your ADC is 1K Ohm and you're using low frequencies so it's relatively accurate. Just be aware that your ADC only sees 1000:1050 of the voltage you're setting on your generator.
Should I be connecting the function generator to the ADC pin with a 50ohm resistor in series?
No ... you could use a 50 Ohm parallel termination at higher frequencies though, you could put your scope on 50 Ohm input if you put a splitter straight on it's BNC input with a very short cable to the ADC, that way you use the scopes internal 50 Ohm termination (not valid at very high frequencies, but with the generator only doing 20 MHz it's okay).
All that said, you should really read up on termination ... because all I said is probably voodoo to you without the necessary context.