Around here usually the specialist first prescribes something, then refills are mostly handled by your GP (unless he refuses to prescribe something for some reason, I don't think he'll do refills on MAOI's for example so my pdoc provided those scripts), it's usually quicker than going back through your specialist. I can log in on the site of my pharmacy, select what scripts I need repeated, then the pharmacy will contact the GP's assistant, who will ask the GP to sign it. And I can pick up the meds a couple of days later.
My pdoc usually provides the benzo scripts (because if you don't indicate on the script that you need them for a certain condition (I think it's epilepsy, pallative sedation, anxiety disorders that haven't responded to a couple of antidepressants, and extremely complex psychiatric problems), insurance won't cover it. So for example if you "just" have a sleep disorder, or take them for performance anxiety if you have to do a presentation or a test, you'll have to pay for them yourself. And the assistant never writes that code on the scripts, so I'd have to pay for them myself, so pdoc prescribes them), as well as new medications, as well as dosage changes.
Pdoc is willing to write the scripts himself, but this is just as easy. Also, if I need three medications, it really sucks to have to write to two or three doctors to get them.

It's also possible to order the refills by phone, but I generally use the pharmacy's website.

Before I had to quit my three main meds in early November (severe side effects that could have become life-threatening.. had to quit three because they didn't know which med was responsible) I had to go to the pharmacy (or I sent my mother - pharmacy is right next to the grocery store she usually shops at) about three times a month.

For a long time I had to go every week because the pharmacists refused to give me more than a week's worth of anything you could OD on (after a suicide attempt). Originally they wanted me to stop by EVERY DAY to pick them up - they are always open on Saturdays for two hours for emergencies only, but they would also be willing to have someone there on Sundays - but my mother talked them down.

(Sort of funny - a couple of times they gave me too much pills. Like I needed 50, and it lists 50 on the label, and they'd give me 5 strips of 14 pills each, probably thinking 1 strip = 10 pills. So whenever I'd find out, I'd send them an email like "You gave me too much pills, 70 instead of 50. I don't think it's necessary to hand them in as I'll be taking them at some point anyway, I'm just letting you know so you aren't missing medication." (Let's not forget that if you hand in pills to the pharmacy, they'll just be destroyed because they can't trust they haven't been tampered with. Of course if it were something reuable, or if they were to ask, I'd return the excess). After this had happened a few times (remember they still wouldn't trust me with more than a week's medication, so I thought it was rather hilarious they handed me too much) they put a note in my file that they had to triple count everything they give me.
Which I thought was even more hilarious. They don't have to triple count my meds - I count them myself when I get home, or if they aren't busy, at the counter once I get them. They should triple count the meds of people who don't check their own meds, or those who aren't honest about it when they get too much. (Don't mean that in a dishonest way - just that some people think "Hey, too many meds, well doesn't really matter, I'll take them at some point anyway" and I'm like "Hey, too many meds, well doesn't really matter, I'll take them at some point anyway, but let me just message the pharmacist so they aren't looking for those 20 pills that they just gave to me and accuse one of their employees of stealing or something"). So people don't take too much meds (if you only have to take meds for 10 days and they give you 14 pills, you will take them too long) and their records are straight.