Taking a Stray Cat to the Vet?

I've bonded with the neighborhood stray cat, Cleo. She's particularly friendly towards me and will follow me around, rubbing up against me all purring, even outside of regular feeding times. My mom is worried about rabies or other diseases, so I always wash my hands after touching her or petting her. But Cleo does let me pet her and once stepped up in my lap while I was sitting on the porch. My mom said if I take her to the vet, I can pet her as much as I'd like, and I get to say that Cleo has been adopted and that I am her owner. But I don't know how to get Cleo into the cat carrier without being scratched or bit. Cats generally hate being shoved into a box. Even my cat will fight me and scratch me up when I try to get her into the cat carrier. And Cleo is still getting used to me and being touched; we've only been bonding since a week before Christmas.

Does anyone have advice on how to get Cleo into the cat carrier without bringing harm to myself or the cat? I know there's a trick to grab the cat by the scruff and it's supposed to immobilize the cat long enough to get her into the carrier, but I've never done that before and I'm not sure if I can do it, although that does seem the easiest way to go considering Cleo is tiny and lightweight compared to the bowling ball of a cat my Evy is.

Help?