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.
「犬の爪きりに病院へ行かないといけない」という状況は、英語で "I need to take my dog to the vet for a nail trim." と表現できます。ここで "vet" は獣医(veterinarian)の略で、"nail trim" は爪を切ることです。
用事があり明日のランチに参加できないことを伝えたい場合には、このフレーズを使い、次のように言うことができます。"I'm sorry, I can't make it to lunch tomorrow; I need to take my dog to the vet for a nail trim."
さらに関連する単語として、次を紹介します:
appointment：予約
obligation：義務、用事
busy：忙しい
日本 
アメリカ合衆国 
