I'm pretty sure I understand how this needs to be done, but I want to make sure so I don't ruin my Arduino or anything else. I have a 120VAC geared motor that I want to be able to switch on and off using the Arduino. If I understand right, I'll need to use a relay. While relays that can be switched with 5V and less than 40mA and will switch 120VAC+ do exist, I don't have one...I have a relay that can be switched with 12V and will switch 120VAC+. What I believe I can do is use the Arduino and a transistor to switch a separate 12V source which will in turn switch the relay, which will power the motor.
Am I understanding this right? The Arduino switches the transistor, allowing the 12V source to switch the relay, which will enable power to the motor. Do I need a special transistor? Right now I have some TN4403's, TN2907's and TN3906. I read that I need a TIP120, I can only assume it's because the ones I have won't handle 12V or something.
Your basic understanding is correct on all counts. The transistor sizing thing is that the switching transistor must have both voltage and maximum current rating well above what the relay coils voltage and current draw are, so that is what drives selecting the best transistor. Bigger then needed transistor is fine but too small or operating right at it's limit is not. So you learn to read the datasheet for specific transistors to see if it's a good fit for what your circuit will use.