The best rule of thumb is to consider all streams contaminated for human drinking purposes. Most parks have a source of safe water available to campers. Whenever possible, use that water or bring water from home. The only really reliable purification process is boiling. Regardless, at sea level boil for at least three minutes and add one minute for every two thousand feet above sea level for relative safety.
Prices vary as do purification methods.
Spend some time checking these out before deciding which one to purchase. Other equipment needed for camping, such as cooking utensils, tent, sleeping bag, matches, biodegradable soap, and the like, remains the same as for family camping. Plan carefully to bring as little as possible. Of course, you are going to need feed, water, and equipment for your horse, too. Don't forget your grooming supplies and insect repellent for both of you.