If you notice any change in the way a red mole looks, book an appointment with your doctor. Any lesion or growth should be looked at when its appearance changes, or when the diagnosis is unknown. Your doctor can rule out serious conditions such as skin cancer, and may decide to do a biopsy.
A cherry angioma will not go away on its own, and it is unlikely to cause you any problems. It may bleed from time to time if it is irritated. A red mole that changes in size, shape or colour is always a cause for concern and should be looked at by your primary care doctor or a dermatologist.
Spider angiomas often occur in pregnant women and in people with liver disease. If you are not sure what a spot is, ask your doctor before you book cosmetic removal.