While I'm far from a kitty nail clipping expert, this is how I got Niko used to me clipping his nails. I'm a first time cat owner, and I read many posts on here before I felt comfy to even attempted it. Pardon ahead of time for the long winded response, I just wanted to be as specific as possible.
1. First, I made it a point to stroke and touch his paws while he was snuggled next to me or while giving scritches an' loves. I took it slow, not wanting to spook him too fast. I kept doing that periodically and casually until he stopped flinching or pulling back his paws when I went to touch them. When he was sure I wasn't going to hurt his toes, I then started handling the pads and seeing how long he'd let me push out the nails (just press down gently on the pad as if you're handling one of those car key remote entry thingies).
2. I then started introducing the clipping instruments, letting him sniff them and rub his scent on them so they didn't seem like foreign contraptions of doom.
I've tried three types of clippers; scissor type, guillotine type, and finally, just a pair of human toenail clippers. He was cool with the scissors, but the cut wasn't very clean. The guillotine just freaked him out, as it was a bit large.. and for me, a little unwieldy. My fav are the toe nail clippers, as they're unimposing and I get a clean cut. I hold he clipper so that the blades are vertical, as opposed to how you would do you're own nails as I find it less likely to 'crush' the nail.
3. I then waited until he was calm and comfy- you don't wanna try to clip the nails of a cat who is in play mode/hyper- it just doesn't work. I found the best time to clip his nails are when he comes looking for affection, as opposed to a random ambush. I layed down beside him and wrapped an arm over top him, so that his rear was tucked in my armpit. I held his paw with that hand, while the other wielded the clippers. I was sure to speak softly and calmly, telling him what a good boy he was. A trick that worked when he was still iffy about it all, was keeping a few treats within the palm of the hand holding the clippers. The moment he'd let me do one nail, I'd let one treat drop. They're also less likely to try to run away if they can smell that there are treats close by.
I only nip the tips off, as opposed to trying to get as far down as I can before getting to the quick, as it's much less 'invasive' for him. Also, any pressure exerted on the nail itself is minimized.
As a side note, while he's now comfortable with me clipping his front paws' nails, he's still not too crazy about me touching his back feet. Luckily, though, he seems to take care of those himself.. I catch him regularly chewing on his back claws, trimming them down. I've managed to clip them two or three times since I've had him, but I noticed that the more I clipped them, the quicker they became razor sharp.. so now, out of preference, I only concentrate on his front paws. Once, I attempted the 'burrito' method in order to get at his back feet, and honestly, it was more stressful than it was worth. This was a year ago, and to this day, the blanket I used to wrap him up is his arch nemesis. He won't even come near it, and if I happen to be carrying it, he runs and hides from me... he NEVER hides from me any other time.
Anyhoo, I hope this helps.