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You need to trim your pet's nails better. Your vet staff cannot do it in the best way, given the limited time & resources they have. Let me help you with this thread. I'm writing this as I go along-- bear with me.
Keep the nail clippers out around the house so they are no big deal. Pick it up, let your pet sniff it, & reward them. Do this often enough that they are obviously happy when you go get them.
Reward your pet for letting you play with their feet in the same way. Desensitize them to handling with a game of Touch-Treat-Touch-Treat, etc.
I could talk more about technique regarding cooperative care training and desensitizing to handling upon request...meanwhile...
All nails have a blood supply in the middle. Don't let clear nails with a pink, visible quick fool you-- the quick tapers off to a point inside the nail and extends further than you think. I actually think black nails are easier because of this-- no deceit.
Take your trimmers and cut off the tip.
I'll add pics of my dogs' nails tomorrow. They are sleeping now.
Look at the end of the freshly cut nail tip-- it is all a uniform color. Clip again.
Look again. Still all one color. Clip again. Look. Once you get to the quick, a clearly visible circle of skin appears on the tip-- pinkish or blackish, depending on the color. STOP. If you go too far, they will bleed. Shave off bits at a time instead of one big cut.
(Use corn starch or Kwik Stop to pack on the end to stop bleeding).
Do this every two weeks or so. The more often you do it, the more the quick recedes so you can get nails shorter and shorter.
Keep nails short enough that you don't hear clicking when they walk, if possible. It is much more comfortable for them.
Just do a couple nails a day so you don't try their patience. If they are done and leave, let them go. Reward them for staying.
Sometimes pretend to cut but don't really, and reward them. If they don't know if you really will trim or not, but they get a reward regardless, they'll wait and see.
Same rules apply for grinding/filing. I never do it because it find it harder to tell when the quick is close-- it is *too* gradual-- and my autistic self loses my shit at the sensation & sound & smell of dremels. People make dogs bleed with them more often, in my experience.
Same for for bunnies, quinea pigs, birds, etc.
A lot of trainers teach dogs to paw at a "scratch board" to file their own nails. I can't handle the sound because of autism, but I think it's cool!
My friend, Leah Fallon ("Hairy Poppins, LLC"), is a new Karen Pryor graduate. Check her out, desensitizing a puppy. (I love watching others trainers-- we are all so different!) https://www.facebook.com/leah.falon/videos/3385894451462591/?extid=zu9vnIxDRmUtC7s2
Declawing cats is the exact same thing as amputating all your fingers at the first joint & taking away part of your language (cats communicate chemically by scratching). It is cruel & often results in lifelong neurological pain.
My favorite style of clippers for dogs:
My favorite for small animals & cats:
Your vet techs are worried about the sick, hospitalized pets &won't have time to mess around. They"ll put on a muzzle &just trim them while another person (or 2 or 3) holds them down &another person gives canned spray cheese as a happy distraction. Everyone hates it.
At least once a week, as a vet tech, I trimmed cat claws or dog nails that were so long, they were curved around and growing into the paw pads. 
Bird nails:
Guinea pigs
Poor rabbits always have nails that are too long. Trim them.
For people w/out help, living alone, who haven't yet trained cooperative care, learn how to SAFELY (many ways aren't safe) wrap smaller pets (like birds & bunnies) in a towel (like a swaddle or burrito) to restrain them. It should be done like a rewarding "game".
Groomers of the world: Do you trim nails like the vet tech way I described, too? Like, are you having to watch the clock and complete everything no matter what?
How to make a scratch board: https://www.facebook.com/watch/?v=1063554654000642&extid=DuDZrx7zid6xPc44
Scratch board part 2: https://www.facebook.com/watch/?v=659049644895003&extid=oRaHtTkNsYfjqOo2
Teaching scratch board use:
Dog Nail Trim: Training a Dog to Enjoy Toenail Trims https://drsophiayin.com/videos/entry/training_a_dog_to_enjoy_toenail_trims/?utm_source=twitter&utm_medium=social&utm_campaign=SocialWarfare
