Remote control for a battery operated train
Power Functions
I understand the train you have is Winter Holiday Train (10254). Luckily, LEGO has already has thought about it and have an option to power up this train. If you still have instructions check the last pages where you can find the list of elements you need (also pictured below) as well as 11 more steps you need to follow to get them installed.
In my opinion, 8879 controller pictured here is way much better than other Power Functions controllers, simply because you only need to set the speed once and train will keep running. Unlike the other types, where you have to keep pushing control toggle to keep your train moving. All of them can be found on Bricklink.

Powered Up
Since Power Functions elements are no longer sold by LEGO you have to look for other means of acquiring them new or used. However this system has been superseded by entirely new system called Powered Up. In some way it is even better for trains since you no longer need a place for bulky IR Receiver. You would need following elements to motorize your train with Powered Up:

Train Motor (set #88011 comes with train wheels, or as an individual element #bb0896c01)

Hub (#88009 | #bb0892c01)

Remote Control (#88010 | #bb0895c01)