Are there programs for free or subsidized birth control?

Yes! Planned parenthood is one program that has been providing discounted birth control, cancer screenings and std screenings for women. Also, check with your local walgreens, walmart, cvs pharmacies. Many of them have begun to offer $10 monthly pills for certain generics. Bring that list to your doctor so she/he can advise you what will work best for you.
Yes. Walmart and most pharmacies offer a list of very low cost birth control pills some for as low as $12 for a 3 month supply. Most cities have public health clinics which do also. Your doctor should be able to give you a list of options.