Your pets getting sick is a very stressful situation, as they cant tell you what is wrong with them. Here is my experience of taking my turtles to the vet.
Some time after changing their tank, I found out that my male turtle is not going into the water or even eating, i was concerned and worried about him, Sometimes he will close his eyes and sleep even when the lights are on during the day. So we decided to bring him to the vet the next day.
I transported him in a small container with a small amount of water. He pooped a lot on the way and is more energetic than at home maybe because of the car's movements. Upon reaching the vet, a veterinarian is assigned to check my turtle. The veterinarian said that it has a bacterial infection or and eye sore and needs some antibacterial injections and medicine that I have to give at home. The veterinarian did those steps effectively, I'm just feeling a bit uncomfortable seeing my turtle getting an injection. The veterinarian did told us the proper steps to feed him the medicine by irritating him or luring him with food.
Even with the advice and steps to feed my turtle, we find it difficult to feed him the medicine at home. Since he is a quiet turtle and not as aggressive as the female one, he does not bite even when we irritate him or lure him with food. sometimes he will just raise his head to avoid it. And when he does not feel any food or medicine nearby he will just close his eyes.