We have dogs and horses at our farm. Lots of both live here and then we get lots of annual visitors many of whom bring their dogs with them.
Our horses are very dog savvy - which is not to say they ignore dogs, some do, some don't.
Some dogs seem naturally sensible around horses, Some are not.
Our first goal here when doing horse/dog introductions is safety. That means I'll ask people to keep their dogs on leash and out of the paddocks. If the dog isn't interested in meeting the horses that doesn't bother me at all for short term visitors but if the dog is going to be a permanent resident it's important to be sure that they are OK together (or not OK so you can implement a management strategy).
Adult to adult introductions may be harder than puppy and foal introductions. Dogs are predators and horses are prey in the global scheme of things so it makes sense that introductions are not always instantly easy. If it's your dog getting used to horses you can ensure some basic training is in place .... "leave it" and "go to place" are two behaviours that can be very helpful. With them you can redirect the dog to it's own space where it can watch the horse with as much distance as is required and slowly build comfort. With new horses we keep them in a stall at first - that way they can view their new life from safety and we get a chance to see if they might be dog wise already. Turning a horse out and having them terrified of a dog can be a recipe for disaster. There truly is no need to rush introductions.
Be positive about the introductions - reward the horse and dog both for calm greetings and interactions. Yelling and punishing either will only serve to increase nerves and distress for all.
If you have a sensible horse and sensible dog you might want to ride with your dog. Please be aware that the first time the dog pops out of a tree line or long grass unexpectedly your horse is simply channeling wise instincts if they spook. Be prepared! Riding with your dog in the ring might be a good idea as preparation so might taking your horse for a hand walk with the dog loose doing it's doggy things. There is no shame in hopping off your horse and walking both dog and horse home if things get a little too exciting for your liking!
The first day to introduce your dog to horses is probably not the day the blacksmith visits the farm. Most dogs love hoof trimmings and can get quite seriously underfoot when they really aren't welcome. They also may be so focused on getting the hoof bits they aren't paying attention and may be more readily hurt. Horses are much bigger than dogs and one kick or strike can be lethal to your canine or result in serious vet bills. Dogs can be loud and frightening to horses and the reverse can also be true - if your dog gets horses running into fencing for example the results can be fatal. be aware of the risks and work with both equine and canine to set yourself up for success.