Working with pets is like working with adults or children in that you, the photographer, have to understand your subject. You've got to go in there confident, calm and with the best methods you have to help control and direct, whilst also preserving a more relaxed and natural appearance of things.
For adults this might be a few jokes, for children it might be sweets or a squeaky frog toy, for pets it might be bacon bits, squeaky toys etc.... General familiarity with animals will help you a lot, of you know how to conduct yourself around dogs, not just your own but others as well, then that will help you a lot. Further when taking photos of other peoples pets, remember you won't be there alone, their owners will most likely be present at the shooting so take full advantage of that fact = position the owner so that the dog looks in the right direction, have them call out or use a favoured toy etc....
Different animals will be very different in how they react and sometimes you might well find you have to take longer in order to get the shot - you've got to play things by their rules.