Since these doorbells are made to work with mechanical chimes, I assume I need to add a resistor into the circuit.
I would assume so... Lighted doorbells and lighted wall switches are powered through the load. When the button is pressed (or the light switched-on), the light goes out.. If we assume 1/2 Amp - 16V @ 1/2 Amp would be 32 Ohms (at 8 Watts). Of course that's just a guess, so the resistor doesn't have to be exactly 32-Ohms, but it does have to be rated for the wattage. (It will only dissipate significant power when the button is being pushed.)
The voltage should "appear" across the load when the button is pushed. (There will be a small voltage at all times because there is some small current required to power the thing, and current current will be flowing.)
But, what is the best way to hook these circuits into the Arduino pins so I can detect the button presses?
Do you have a multimeter?
That's AC, and the negative-half of the cycle can kill an Arduino. so, you'll need a rectifier (diode). You'll also need a [u]Voltave Divider[/u] (2 resistors) to bring tgeh voltage down to 5V. The resistor values are not critical, it's the ratio that's important, but if you make them add-up to about 10K that should work.
Note that the peak of an AC waveform is about 1.4 times the RMS value, so 16VAC will peak at about 23V. And, the transformer will have a tolerance and the voltage will vary with the actual power line voltage, and with the load... To be safe, you should add a [u]Protection Diode[/u] along with the voltage divider. (The resistor is an important part of the protection circuit, but the resistors that make-up your voltage divider will suffice, so you just need a diode.)
Since the AC is transformer-isolated from "everything else", you can connect either end of the resistor to the Arduino ground and the other end is the "signal" through the rectifier & voltage divider to the Arduino's input. Note the "reverse logic"... The Arduino input goes low when the button is pushed.