Okay, on my game all the animals pee all over at first. You have to SCOLD them. When you are scolding them a little learning bar comes up over there head and each time you catch them red-pawed you have to scold them. When you've scolded them so many times, the bar will fill up blue and a message will say your pet has learnt not to pee, rock on. (this happens with chewing too, scratching etc)
Then, make sure you have a shrub or tree, maybe even flowers and the pet will use this to pee on.(DOGS) and cats will need a latrine (from catalogue or pet emporium)