Good on you for helping these kitties out
Are you feeding them? If not, start feeding them at the same place, at around the same time each day in the evening preferably.
Once they're coming regularly, stop feeding them, then set the trap out the next night, so they've had 2 nights of no food. I find the best time is at dusk.
Put something nice and smelly in the trap - tuna, sardines, BBQ chicken, and put a towel over the top, but don't cover the end of the trap - they're more likely to go in if they think they're entering a "tunnel" rather than an enclosed space.
Keep an eye on the trap because once the kitty is trapped it's going to flip out. Have a sheet, or really big towel nerby that you can put over the top of the cage. As soon as it's completely covered they settle right down.
Once you have it, just put it in a bathroom, a warm garage or something else like that where it won't get disturbed or too cold.
The kitty will be fine in the trap overnight in a warm place, just make sure it's completely covered, and then leave it be. Don't give it any food or water, and don't go trying to stick your fingers in the cage, because they are going to want to bite you
When you bring it back from the vet, just keep it in the trap in a warm place. I wait about 8 hours after surgery, then put a bit of food in the cage. Don't open the trap too far, open it just enough to slide some food in.
I generally keep the cat one night, and release it the next evening. If it's female, and it's doing fine, I keep it 2 nights. I have a feral cat recovery cage, so they have a litter tray and a bit of rooom to move. If I didn't have that, I would probably only keep it one night.
If it's a female and in heat or pregnant, you'll probably need to keep it a bit longer, so talk to the vet about the best thing to do.