A rule of thumb cited by the Talmud is: What comes from a kosher animal is kosher; what comes from a non-kosher animal is not kosher.
Thus, only milk of kosher mammals is kosher. Additional kashrut provisions pertaining to milk are discussed in Separating Meat & Dairy and Chalav Yisrael.
The same applies to eggs: only eggs laid by kosher birds are kosher. In addition, all eggs should be carefully examined before use to ensure that they are free of blood spots.
Honey is not considered an "animal product," so honey is kosher though bees aren't.