In the first place, a door bell has a simple logic in it. If you press the button it would ring, otherwise it won't. In the most simplest for that is the function of the OR gate. But if you want to complicate things that is up to you.
It is vital that you understand how Logic gates work. They are just switch to put it simply. But it depends on the output you want to get.
I would suggest a push button switch in series with the door bell circuit if you want to achieve an on or off operation. I don't see the need of a logic gate. If you really have to implement your design using a logic gate then by using an AND gate with two inputs, you can just make one of the input pins logic high and connect a push button switch to the second pin, in essence the output will follow the input of the second pin which will then be connected to the door bell circuit. But as I said earlier, the second design will just be a complicated version of the first design
