Make sure the pliers are easy to open and close. The new pliers which I've seen have all been stiff. Work them open and closed to loosen them up, oil them if necessary. One of the jaws on the pliers has kinda of a little 'hook'. Think of a little werewolf tooth poking up. When the pliers is held in the correct orientation for tagging, this is on the Bottom.
Inspect your tags. If they came through the mail, sometimes they can be squished by the postal service. Load a tag into the pliers. Very slowly close the pliers, until you can see the point on the bottom of the tag approaching the hole on the top. Are they about to meet nicely? If the tag is warped then the piercing bottom part will not go cleanly through the hole on the top. On a cavy ear this would likely clamp onto the ear, rather than piercing the ear and bending around to secure itself through the hole. If the tag is warped, it can easily be bent back to the correct shape. Properly positioned, a cavy tag does not pinch a cavy's ear in any way, as it is designed to rest there on the ear hanging from the tiny hole it has created when you crimped it on.
Select the site for the tag. Study the position of the tags of other cavies you own. Are they on the forward part of the ear? Closer to the cavy's head than the floor? This prevents most cavies from accidentally getting a scratching toenail caught in the tag. Are they too high forward? I acquired a cavy whose tag was so far up the ear, it was virtually in the cavy's scalp. Plus it's not so easy to read a tag obscured by the hairs there. One advantage to using a high position on the ear is if the tag does rip out, there remains plenty of space below it for another.
How far into the ear to tag? Who has seen tags biting into the ear's flesh because the animal was tagged young with no space for ear growth? This takes some guestimating-- for older cavies don't worry about leaving any empty space between the edge of the ear and the end of the tag. For a young cavy, leave no more than about an eigth of an inch. Probably less. If too much space is left, the tag becomes a protruding hazard that will be scratched out or pulled off by an inquisitive cagemate. For any tag applied with a little space, gravity will swing the tag toward the floor and the inside of the tag should rest against the edge of the ear. For newly inserted tags this might not occur yet and you can manually manipulate the tag to do so (the freshly tagged cavy might complain).
When tagging a hemmed ear, be sure to study the ear and unfold the part you intend to insert the tag into. This prevents you from getting the tag too far into the ear, with two thinknesses of ear crammed up into the tag, or the tag inadvertantly punching through two ear layers (for those heavily hemmed ears!)
What is the cavy doing during this time? Probably becoming impatient at the least and at the worse, squealing and jumping. I always hold the cavy against my body while tagging. Some people like to have an assistant hold the cavy. Do whatever is more comfortable for you. With the cavy's body held against me by my forearm, I use that same hand to secure the cavies head. You want to be sure the cavy neither jerks backward nor jumps forward. It is handy for me to partially close the tag ahead of time, but not mandatory. If you overclose a tag and cannot get it over the ear, you can easily pry it back open. It's nice to have a pliers that swings open again immediately as you release it. Work it and oil it if it doesn't. I saw an ingenious fancier place a spring device into the handles of their tagging pliers, similar to the coiled metal found in single hole punches. Of course the pliers spring back open immediately. This is for the cavy that is really fighting you and there's a danger to having the tag rip out of the ear before you can release the pliers.
You may wish to tag a few "non show" cavies first. Practice makes perfect. The first time I tagged, I was so nervous. I had borrowed someone's pliers at a show and imagined that everyone there was watching me. They weren't. I applied the tag upside down!!! Some tags spin around in the ear so it made little different but still! My first time and I 'screwed up'.
We used to put campho phenique on the ear, as an antiseptic and anethestizing agent. But I read that ingredients in there weren't good for cavies. (camphor, phenol). Not sure of the truth to that, but I found no difference in using an antiseptic on the ear or not. Cavies either squeaked when the tag was applied, or said nothing and seemed to not notice. I thought perhaps by wetting the ear with something I would actually encourage dust and the like to stick to the ear so I discontinued putting anything. Most people don't. The ear almost never bleeds. There don't seem to be a lot of large vessels in that area. Rarely one will bleed. My luck it is always a white! Just pinch the area (difficult to do, there's now a piece of metal in the way) until bleeding slows or stops.