A piece of information: the dog appear after 10 days in the town, you need to feed it at least 5 times
Build the doghouse (Idk why but my doghouse icon is a horse, maybe because I also built the stable), there you can train your dog/horse/raptor.
You can teach your dog how to sit, help, mount you and search, but for those it will take energy so make sure you are constantly feeding him while you train him.
P.S. If you won't have as much options as me, I also have an explanation (kinda, I think?): some time ago I've read a comment that said if you tease, do dirty things (you get it what I meant) with the wolves in the forest you can learn more tricks for your dog.
P.P.S. I do not know if this theory is real because when I read that comment my character already had a bitch reputation and I also had the dog trained to do all the staff I mention.
Hope this will help you, let me know!