You shouldn't take them out to public places till they have all their shots - but you know that. So, I started getting Rowdy and Rexy used to their harnesses at about 11 weeks (put it on them in the house) so when they had their shots at 12 weeks, then they were ready to go and used to having the harness on. A few days after the final shots I start leash training. Rexy was decidedly easier than Rowdy - becaue Rexy just follows Rowdy - kinda like Pete and RePete!
