I always have pain with mammograms. Should I ask for a sonogram instead?

No. Sorry but please get your mammo. Try different technologist.
No. An ultrasound is not an alternative to mammography. Speak with the mammography staff and see what you can do to limit the discomfort. They likely see people with the same problem and may have advice for you.
No. Mammograghy is still the best procedure we have for detecting early breast cancer in our population. Ultrasound and breast MRI are very useful in the appropriate setting but are not the best for screening and cannot "replace" an annual mammogram. Unfortunately compression of the breast is necessary for optimum images and can be uncomfortable for some.