Q:Regarding the rule always eat a carb with a protein, is milk considered a carb or protein or both?
Milk contains both whey protein (fast acting) and casein (slow acting) as well as carbohydrates, an average of 12 grams per serving, at least 11 coming from sugar (from lactose in the milk). A short answer is that milk is both. It is important to be aware of your goals when you drink milk and think about what you are eating with it.
Milk and cookies are the traditional combination but may not be the best if your goal is fat loss. Milk has at least 11 grams of sugar per serving, so combining it with a cookie (sugar and fat) is probably going to leave your body with more carbohydrates (energy) than you will be using. The same thing really goes for cereal. Even moderate sugar cereal has about 7-8 grams. If you are careful to keep the milk serving to 1 cup and the cereal serving measured (which few people do), you are still looking at 18-20 grams of sugar for a bowl of cereal.
It is really personal preference whether to drink milk or whether to combine it with other foods. When making smoothies, you can choose to use milk, or you can substitute plain Greek Yogurt (which as comparable protein but less sugar) or unsweetened vanilla almond milk or unsweetened soy milk. The almond milk does not give you the same protein benefit, but it will significantly decrease the sugar content of your smoothie. Adding a NON GMO protein powder can give you the protein benefit as well.
Some have suggested that chocolate milk is a good post-workout drink. Keep in mind that a serving of chocolate milk has about 27 grams of carbohydrate (25 coming from sugar). If you make this choice, be aware of the significant increase in carbohydrates to determine if your workout was intense enough to justify the addition of chocolate to your milk. Moderation and education are a great balance. I rarely use the words always and never when it comes to food choices. Just being aware of nutritional breakdown can help you make choices that help you achieve your goals.