@umbra21 - With a bichon frise puppy that shouldn't be a problem though. They don't grow the longer coat until they reach about a year old.
And remember, if you live in a place with low temperatures, you need to make sure your puppy, or even your adult dog will stay warm. If you cut off the entire top coat, it reduces their ability to stay warm to even lower than a short coated dog, because you remove all the guard hairs. Which can be a good thing in a hot climate, but very dangerous if you live in a place where it snows and the dog is often outside.
It's a very good idea to go to a professional groomer in your area, at least once, and see what they recommend. It might be different styles for different seasons. Please go with what will make your dog the most comfortable, rather than trying to be fashionable.