Based on what happens in my yard, it should work to plant the pad but not the fruit (you might be able to germinate seeds from the fruit, but you'd wait a long time for them to grow big enough to bear any fruit of their own). However, unless you're certain that the pads they're selling are of the same variety as the fruit, then there wouldn't be much point in planting the pads--you'd be better off to find another source where you know you're getting a variety that produces good fruit.
Our native prickly pears (as well as some non-native Opuntias) seem to do best with plenty of neglect. If I pull up a plant and leave it lying on the ground, or even drop just a single pad on the ground, it will almost always root in and start growing during the summer rains. But--if you break off a pad and lovingly plant it without first letting the "break zone" dry out and callous over, it will often rot. And if you plant it, and then immediately water it without letting it callous first, it will almost certainly rot.