Try to think of it like this: We use doors to enter rooms (generally speaking). We must pass 'through' the door first in order to go into the room. We don't go 'in' doors. We go in/into rooms. Think of through as means of passage. It is a continuous action as opposed to in, which is static. For example: The train went through the tunnel. (train went in and came out) The train went in the tunnel and crashed. (train went in, but didn't come out - it ended up motionless) Did the burglar enter the house through a window or a door? He entered through the back door. Through in your example is the best option. You can say 'You can go in any door' and people will probably know what you mean, but grammatically speaking, it isn't correct (though lots of people use it). I hope this makes it a little more clear for you.

Using "through" is probably better. Would not use "on".