As a note, you do not need a "Range Membership", many of which cost several hundred dollars per year. You need a "Club Membership", many of which cost less than a hundred dollars a year. For example, I am a member of the Edmonton Shooters Guild, which costs $30 a year, but I also have a membership at the Phoenix Indoor range which costs $395 per year, huge difference. The range is optional, the club is not.
Once you have that, whoever currently owns the pistol will need your information, such as full legal name, address, PAL number, and so forth. They can initiate the transfer with the CFC, who will then need you to confirm the transfer.
Then, if you will need to pick up the gun at the store, or post office, or whatever you will need an Authorization to Transport for that, from wherever the gun lives now to your residence. If the gun will be delivered directly to your door via courier, you do not need an ATT for that because you're not transporting it anywhere.
Then, if you want to take the gun to the range, or gunsmith, or wherever else, you will need a different Authorization to Transport from your residence to that destination and back again.
Now, since you called it an FAC, I'm just going to mention that you should make sure that your contact information with the CFC is up to date. They will need your current address. Make sure that is all updated first, so that it doesn't hold up your transfer if they're trying to call some number that's now out of service, or mailing notifications to a house that you moved out of back in 1983. This is a hole that a lot of older guys are falling into, especially if you have been moving around with restricted or prohibited firearms, you need an ATT to change residence and many people forget about that. If you call in to acquire a new gun, but provide a new address that the CFO doesn't have on file, you can get in some trouble if they find out you've been illegally transporting your firearms.
Also, the CFC, and CFO are now able to email notifications, which can significantly reduce wait times. In many cases you can get your request processed in minutes rather than weeks spent waiting for paperwork. Some documents, such as the registration certificates will need to be mailed, but the sender can send the gun as soon as it is approved and then you just sit on the gun until the certificate arrives. Usually much quicker that way.
If you do get things like your ATT emailed, I strongly recommend printing off several copies. Leave one at home, keep one with the gun, and keep one in your pocket. If all you have is a digital copy on your phone, and the police ever need to examine it, you're going to have to give them access to your phone and/or email account so they can look at it. They aren't all like it, but some officers will walk away with your documents and sit in their cruiser, and then additional officers will show up and demand your documentation and saying "The other officer has it." just sounds like an excuse they've heard a hundred times.
Some people will recommend claiming that you are a collector as a way to "get around" having to buy a club membership. Don't do that because the CFO may decide to deny you a long term ATT to go shooting at the range because you chose collecting rather than target shooting. Also, it opens you up to having to meet the collector's regulations.
Enjoy.