I you are going to be taking Luigi to a groomer I would suggest that you wait until your puppy has received all of his shots. I took Roxie to get groomed for the first time when she was 4 months old. All of the groomers in my area require shots as well as the bordetella vaccine every 6 months.
I would wait until Lucy gets over her kennel cough before getting her groomed. I hope she is getting better on the antibiotics. My little Roxie got kennel cough a month ago. I am not sure if the kennel cough was from the bordetella vaccine or the groomer. I think that it is great that you found a groomer that will come to your house. They look adorable after their first hair cut!!
