If buying arugula in bunches, cut the bottom of the arugula off and discard. Then separate the leaves and give your arugula a good wash in the sink as they hold a lot of grit. By swishing it around in the sink, it will let the grit sink to the bottom of the sink. To dry, you can use a salad spinner or pat dry with a towel.
Arugula makes a great salad – try it with goat cheese and an olive oil based dressing. Also, try adding a handful of arugula to a veggie scramble, to a pizza just out of the oven, or try wilting arugula and adding it to a panini with pancetta and cheese.