How do you say "train" as in " I train the dog (or horse)"? How about "The mouse was trained to avoid cheese" I am not looking for the word for "teach".

to train → antrene, displinen, fòme, prepare, anseye, edike, fè mache ès-ès

Since 'antrene' is best used as an intransitive verb, your best choice here would be 'disiplinen'
I trained the dog to use the litterbox..
Mwen disiplinen chen an pou'l itilize bwat lityè a.

The mouse was trained to avoid cheese.
Yo te disiplinen sourit la pou'l refize fwomaj.

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