How old should a woman be to get a first mammogram?

It depends. For most women, age 40. For women with a history of breast cancer in a mother, sister, or daughter, possibly younger. This depends on the age the relatives developed breast cancer. Ultrasound and MRI are better tests in young women with dense breasts. For women with a suspicious lump, a mammogram, ultrasound, and possibly breast MRI should be done regardless of age.
Age 40, unless. If you are a BRCA carrier , or otherwise of very high risk, it may be appropriate to begin screening early. If you have a lump or other symptom, see your doctor, ultrasound would be the first imaging test of choice in your age group. Otherwise, the American College of Radiology, American Cancer Society, and others recommend screening starting at age 40.
Between 35-40. In the abscence of significant risk factors, a baseline mammogram can be obtained between 35 to 40 years old. Thereafter it is recommended to do yearly mammography after age 40.