Here in the UK the doctor prints out the prescription on special prescription paper and gives it to you. It looks like this:
Prescriptions must have a doctor's 'stamp' on them containing the surgery's name and address to prove they're genuine. This used to be an actual rubber stamp, but now is computerised, mostly. It also had to be signed by the GP, although I'm not sure if that's still a rule with the electronic-ness of the stamp now. The doctor then gives you the prescription and you take it to whatever chemist you like - there are usually a few independents plus chains like Boots or Superdrug on every high street, a lot of big supermarkets have a pharmacy dept, many small local shopping areas on housing estates and the like have an independent, and there'll usually be one close to the GP's surgery - and wait while it's prepared. It doesn't usually take any more than 10 minutes even if there are a few people in front of you.
Prescription charges are £7-something per item in England but free in other parts of the UK. People on certain benefits don't have to pay for their prescriptons and must tick the clause that applies to them on the back of the prescription and sign it before handing it to the chemist. People with certain chronic conditions like epilepsy also don't have to pay, and have what's called an 'exemption certificate'. So, there's no checking of insurance details on the computer when you take it in because you either tick or pay. Even if you do have medical insurance and see a private doctor, I think they'd still give you the same prescription and you'd take it to the chemist and pay the prescription charge, but don't quote me on that because I don't have it.
Many chemists operate a delivery service for people who are elderly or mobility impaired and can't get out to pick up their prescriptions. In that case, the GP's surgery will send all the prescriptions that need to be delivered to the chemist (usually the one closest to the GP's surgery who deals with them all) and there will be a delivery round sometime usually late afternoon. It's a really good service and doesn't cost anything to the patient. My elderly father uses that service all the time and I've used it on occasions when my mobility has been too bad to get out.
On my last GP's appt, the doctor told me that they've just started sending prescriptions electronically; you have to have a 'nominated' chemist on your medical records and then it'll automatically go there and you can pick it up. I prefer to have it in my hand though.
In the picture, that's a 'repeat prescription', for meds that you have to take all the time. The white piece on the right is the repeat slip, and you retain that when you give the script to the chemist. When you're about to run out you tick what you need, drop it into your surgery and they'll do you a prescription to pick up. You usually have to give 2 days notice, ie, drop the repeat slip back and your script will be ready to collect a couple of days later.