Maybe your teacher had tried to say that no need to add 'that' with 'besides'.It's a redundant word in most cases. But I never thought that it would be a serious blunder to add it.

I am not an English teacher but a native english speaker and here is how I would use those words. 'Besides going to the beach I am not going out today' meaning that other than or except going to the beach you will not be leaving home that day. Besides that can be used in response to a statement or question. For example someone may ask 'Are you going to the beach' and you could reply ' Yes but besides that I am not going out today' so you use 'besides that to refer to the earlier question. Hope this helps