Hello Kyle,
a transformer is the best solution in this case to first lower the 120 V mains to a safe level. I don't know about the door, but the keypad, RFID reader and arduino definitely will need DC power. So after the transformer you'll need a rectifier, smoothing capacitor and probably a voltage regulator for each voltage.
You have basically two options:
1) use a 120 V/ 18 V transformer. Rectify and smooth to ~25 V DC (that 18 V * 1.414 because the AC peak voltage is higher than 18 V). Add three voltage regulators to generate the required voltages.
2) use a transformer with two taps: 18 V, 10 V. add 2 rectifiers and 2 smoothing capacitors. Place the 22 C regulator after the 18 V tap, the 12 V and 9 V regulators after the 10 V tap. This will generate less waste heat in the 12 V and 9 V regulators.
In both cases you gave to take into account the power requirements of your circuits. Each voltage regulator (and associated transformer winding) must be rated for at least the nominal current you are going to draw from it.
Anyway, if you are not well versed in building power supplies, use three appropriately specified AC adapters ("wall warts") and leave the safety relevant handling of the high voltage (120 V mains) to the experts. This is what I'd suggest.
Whatever you do: keep safety in mind.
Harald
