I have to go to the vet and have my dog's nails clipped.
I have to get my dog's nails clipped at the doctor's.
I have an errand to run.
Hospital for animals is called "vet."
If your dog is young, you can also say, "my puppy."
When you have an appointment or something that you have to do, you can simply say, "I have an errand to run." This is like, 『ちょっと用事があって...。』which indicates you have something to do but it is nothing serious.