I have a rear rack and a pair of Sunlite Grocery Getter panniers. The Grocery Getters are tough and sized to fit a grocery bag in each.
One thing you will need to learn is to pack your panniers so that each is approximately the same weight, so its balanced. The easiest way is to select two things of equal weight at one time and put one in each pannier. It's not an exact science. And I tend to shop more frequently.
The panniers also provide a base that you can use to bungee extra items to your rack.