Consider walking around the south side of the walls around the Vatican, seeing the relationship of the piazza to that area... There's not much north of the Vatican except a touristy shopping district...
Takes at least half a day if not longer. But ABSOLUTELY worth it. Make sure you do a guided tour. Well worth the price. We did Maximus Tours (good price, great guide): Takes at least half a day if not longer. But ABSOLUTELY worth it. Make sure you do a guided tour. Well worth the price. We did Maximus Tours (good price, great guide): http://www.vaticantou...miprivate/7…see moresee less
Navigating the area around the fountain can be confusing...finding it, as if you were lost and unexpectedly came upon it is most satisfying... Maybe explore the area rather than explicitly planning on seeing it?