General Guidelines
In general, only breastmilk or formula should be used if your baby is less than six months old.
Between six and twelve months, supplementing with solids (instead of formula) or very small amounts of cow, goat, soy or rice milk is less of a problem, as long as baby is still nursing for the majority of milk intake and baby is not allergic. However, babies under a year are more at risk for allergic reactions (see below) so it can be a good idea to wait.
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After a year, other milks may be used, but are not needed (other sources provide the same nutrients). It’s recommended that you limit the amount of cow’s milk that your child receives (possibly other milks too, except breastmilk) to 2-3 cups (16-24 ounces) per day. Too much cow’s milk in a child’s diet can put him at risk for iron-deficiency anemia (because milk can interfere with the absorption of iron) and decrease the child’s desire for other foods. More here on cow’s milk after a year.
Cow’s Milk
Use of cow’s milk before a year is controversial among experts. You might want to get your baby’s doctor’s opinion on this.
The American Academy of Pediatrics recommends NO cow’s milk until after the first birthday.
Cow’s milk is more specific to a baby cow than a baby human. Cow’s milk formula is based on cow’s milk but has been engineered to be closer to human milk (still a ways off, but closer). Many infants still have problems with cow’s milk formula (allergies, GI problems, etc.). Babies who are exposed to cow’s milk before their first birthday are more likely to be anemic, have diarrhea or vomiting, and/or experience an allergic reaction (the proteins in milk are more numerous than those in other milk products, such as the yogurt). The excessive protein load in cow’s milk can also overload a baby’s kidneys. It is deficient in vitamins C, E, and copper. It is harder to digest as well, often causing intestinal blood loss. A number of studies have also indicated that early introduction of cow’s milk may contribute to the development of Insulin-Dependent Diabetes Mellitus.
Others see no problem with starting cow’s milk toward the end of the first year unless there is a family history of allergy to it. Dr. Jack Newman, a renowned expert in the lactation field, is one of these. See the “Breastmilk, cow’s milk, formula, outside work and bottles” section of his article Breastfeeding and Other Foods. Notice that he qualifies this advice with the importance of the baby being well-established on a variety of solids and continuing to be breastfed (both of which should help make up for what cow’s milk can’t offer at this age). He also suggests that supplemental milk of any kind is not all that necessary if the baby is allowed to nurse frequently when with mom. He suggests offering the baby solid foods with some water or small amounts of juice instead.
Some experts consider yogurt and cheese to be okay for most babies after 6 months. Others prefer waiting until 9-12 months. Of course, if baby has a cow’s milk allergy or there is a strong history of allergy to cow’s milk in the family, yogurt, cheese and all other foods made with cow’s milk should be avoided until 12 months or later. The main difference between yogurt and milk is that the lactose in yogurt has been converted into lactic acid. Cheese differs from milk in two ways: the whey proteins are drained off in the cheese-making process (so only casein proteins remain), and the curdling enzymes break down the casein proteins into smaller protein molecules. For babies and toddlers, try to find the whole milk yogurt (not low or no fat), and avoid the yogurt with artificial sweeteners lots of sugary fruits.
Additional Information on Cow’s Milk
Michaelsen KF. Cows’ milk in complementary feeding. Pediatrics. 2000 Nov;106(5):1302-3.
American Academy of Pediatrics Committee on Nutrition: The use of whole cow’s milk in infancy. Pediatrics. 1992 Jun;89(6 Pt 1):1105-9.
Breastfeeding and Other Foods by Jack Newman, MD, FRCPC
Milk & Gruel May Be Cruel by Lynn M. Johnson. “Giving cow or goat’s milk and solids too soon is unhealthy for tiny tummies.”
Cow’s milk for nursing toddlers? @ 
also discusses other sources for the nutrients in cow’s milk
Calcium @ Lots more info here on cow’s milk and substitutes
see also the Nutrition Comparisons below
Goat’s Milk
Using goat’s milk before 6 months or regular use between 6 and 12 months is not recommended. Goat’s milk is no more appropriate to give baby than cow’s milk. If you need to supplement and breastmilk is not available, formulas are a more nutritionally complete product. There are several comparisons of goat vs. cow vs. human milk in the links below. Using this information, goat milk is much closer in composition to cow milk than human milk. Goat’s milk is high in sodium (like cow’s milk) and is very high in chloride and potassium, which makes the renal solute load too high for babies. This can cause gastrointestinal bleeding and can result in anemia and poor growth (these problems are usually undetected until months later). Goat milk is also deficient in folic acid, which can lead to megaloblastic anemia. Also, infants who are allergic to cow’s milk protein are often allergic to goat’s milk too.
While it’s true that whole goats milk (and whole cow’s milk) was commonly used prior to the advent of infant formulas it is also true that the infant mortality and morbidity rate during the times of such substitutions was very high.
Additional Information on Goat’s Milk
Q&As
What About Goat’s Milk? by Lynn M. Johnson. “Goat milk is touted as having exceptional nutritional value over cow’s milk, but for tiny tummies it just doesn’t measure up.”
Goats Milk Formula for Babies from LiveStrong.com
Got Goat’s Milk? from AskDrSears.com (just keep in mind that he is advertising a particular brand of goat’s milk here)
Basnet S, Schneider M, Gazit A, Mander G, Doctor A. Fresh Goat’s Milk for Infants: Myths and Realities–A Review. Pediatrics. 2010 Apr;125(4):e973-7.
Pessler F, Nejat M. Anaphylactic reaction to goat’s milk in a cow’s milk-allergic infant. Pediatr Allergy Immunol. 2004 Apr;15(2):183-5.
Hendriksz CJ, Walter JH. Feeding infants with undiluted goat’s milk can mimic tyrosinaemia type 1. Acta Paediatr. 2004 Apr;93(4):552-3.
Infante Pina D, Tormo Carnice R, Conde Zandueta M. Use of goat’s milk in patients with cow’s milk allergy. An Pediatr (Barc). 2003 Aug;59(2):138-42.
Nutrition comparisons
Why Goat Milk? by George F. W. Haenlein
Goat Milk Formula
Goat’s Milk Formula from AskDrSears.com (just keep in mind that he is advertising particular brand of goat’s milk here)
Dairy Goat Co-operative (N.Z.) Ltd New Zealand manufacturer of goat milk formula
Bambinchen goat milk formula German manufacturer of goat milk formula