What is the process of getting prescription lenses?

You need to get an appointment with an eye doctor and he or she will examine your eyes and write out a prescription for you. You can then take that prescription to a place that makes lenses and ask them to fit the lenses in the frames you already have.
Sometimes you can have the lenses made at the doctor’s office, but it seems to me that lenses are more expensive there.

It takes a couple of weeks for your lenses to be made once you show the prescription to the “lense-maker” (lethologica).

Find an optician nearby. He will do the exam, write the RX and have the lens made so that they fit the frames you own.

In the USA, there are three professionals you can consult for eye care:
An Ophthalmologist is a physician (M.D. or D.O.) with a medical speciality in eye conditions, and they treat medical conditions with surgery, medicine, or therapeutic techniques.
An Optometrist has a clinical doctorate (O.D.) and specializes in vision and visual processing, and can prescribe corrective lenses and also therapeutic devices and techniques. In some states, they can also write medication prescriptions to treat infections and such.
An Optician is licensed in the prescription and fitting of basic corrective lenses only. If they suspect there may be an underlying medical problem, they should refer you to one of the above specialists.

If you have never had your eyes examined before, I would recommend seeing an Ophthalmologist or Optometrist to get a more thorough baseline examination – especially if you have any other symptoms that could be eye-related (e.g. double-vision, sensitivity to light, headaches, dark spots or light flashes, pain or pressure in or around the eyes, imbalance, or perceptual concerns). An Ophthalmologist exam is considered a medical specialist by health insurance – check with your insurer to determine whether your particular policy would cover such a visit, or if you would need a referral from your Primary Care Physician. An Optometrist exam might be under medical specialist, or it might be covered by your vision care rider – again, check with your insurance carrier(s).

If you have no health insurance, and your main concern is just blurred vision at a distance and/or when reading, then call around to opticians and eye care centers in your area to inquire about pricing and guarantees included with the purchase of lenses from them. Explain that you already have frames, and ask what their policy is about using someone else’s frames. Understand that people’s heads are shaped differently, as is the placement of their eyes, so the frames you have may not be suited to your needs.