How would a using a resistor and regulator together affect a circuit?
The idea with a voltage regulator is that the output voltage will be constant no matter what the load is - up to the rated current of the regulator.
With 1 kΩ connected to the output of your 5 V regulator the current will be 5/1000 = 5 mA and since this is well within the current and power rating of your regulator the voltage will remain at 5 V.

Is there a way to calculate this?

Yes. Read the datasheet to work out what the maximum current the device can handle is.
You also need to watch the temperature rise. The regulator will &quot;drop&quot; voltage across it and this drop will be equal to \$V_{IN} - V_{OUT}\$. We can calculate the power dissipated from \$ P = VI \$ where \$V\$ is the voltage drop. So let's say you had 9 V in, 5 V out and were passing 100 mA then \$ P = (9-5)\times 0.1 = 0.4 \ \text W\$. If the regulator gets too hot it will go into thermal shutdown. You can improve the heat dissipation with a heatsink.