Almost. You also need your "pupillary distance." This is basically your eye spacing, which is needed to decide how to space the optical centers of the lenses. Sometimes this is given as two numbers, one for distance, and one for close work. Normally this info is not part of the prescription, but you can ask them to measure it or look on the ever-reliable internet for ways to measure it yourself.
Fortunately, the Chinese are destroying the Luxottica monopoly and also destroying margins for opticians' shops. In addition to frames and lenses prices going down, they can no longer charge for add-ons like $100 for AR coating that costs them $5.
I have been buying my lenses from US shops that advertise "re-lensing" because I like my frames. I use US for speed and (hopefully) better quality control.
Once I have a new prescription, though, I go into high gear on the internet glasses shopping. Typically I can find single-vision glasses with frames on sale at two for $25. (!) I buy enough close-prescription glasses that I can leave one by every chair where I typically read, plus one to carry when I travel. I buy enough mid-distance (half the close distance correction) glasses to have one at every computer screen I use. I have zero interest in the fashion-worthiness of the frames, which is why I can buy on sale, but I have standardized on predominantly black for the close distance glasses frames and predominantly brown or gold for the mid-distance glasses. This keeps me from accidentally mixing them up. The prescriptions are close enough that sometimes I don't notice the difference.
I'm sure I have been doing this for ten years, maybe longer. Vendors I have used:
JEMOptical
Optical4Less
EyeBuyDirect
Goggles4U
LensesRX
replacementlensexpress
I am not necessarily recommending any of these but from my email file these are the ones I have used. I don't recall ever having a problem with any of them.
Here is a web site that is no longer maintained but still has worthwhile information and links:
GlassyEyes | Find Your Next Affordable Eyeglasses Online