Rashes are quite common in the young infant and, fortunately, most of them are more annoying than worrisome. Heat rashes, cradle cap (seborrhea) eczema, and infantile acne are all common, but none are emergencies.
The following types of rashes do need immediate attention, however. The first is any rash that may represent a serious allergic reaction. Though not that common in very young infants, hives (pink, itchy, blotchy spots that come and go in different locations) shouldn't be ignored, as they signal a reaction that can progress to swelling of the face, hands, mouth, or throat and may ultimately interfere with breathing.
A second type of rash that needs attention is one that doesn't "blanch," meaning gently pulling the skin on either side of the rash in opposite directions doesn't make the pink, red, or bluish color of the rash disappear. This may represent leaking blood from the vessels from a serious cause.
Finally, any area of the skin that seems infected—hot, tender, swollen, with or without area of pus—regardless of whether it was preceded by another type of rash, needs to be evaluated.
True vomiting in an infant is serious, but this can be confused with regurgitation or reflux. Many infants, due to the immaturity of their digestive systems, have some degree of reflux, where recently consumed milk comes back up through the mouth or nose. Often called "spitting," it is painless, happens without warning, and is more "gentle" than most episodes of vomiting. It should never consist of blood or bile or anything more than the milk just taken in.
Any vomiting that is forceful or that contains the green substance called bile or any blood is a potential emergency in a very young infant and needs an immediate evaluation. The same is true for any vomiting or spitting that is frequent enough to bring up a concern for dehydration.
Again I stress, when in doubt, call your doctor. Parenting is a situation of lifelong learning, and it is better not to learn the hard way when it comes to your infant's health.