Lemongrass is an essential ingredient in Southeast Asian cooking. Select fresh, large stalks with no bruising and minimal dry leaves. Select tender small stems for making salads and larger thicker stems for flavoring soups and marinades.
Wash well under cold running water. Cut off bottom and top ends and peel away tough outer leaves. Discard.

Your cleaned lemongrass should look like this:

For flavoring soups
Lay the cleaned lemongrass on a sturdy cutting board. With the blunt end of a heavy knife or cleaver, starting at the top end smash each lemongrass stalk crosswise at one inch intervals until it is well bruised. Stop at about three inches from the hard bottom ends. Cut bottom ends off.

Split the hard bottom ends in half lengthwise then smash crosswise until bruised but not splintered. Bruising the lemongrass stalks this way releases their essential oils and fragrance and maximizes flavor. Cut the remainder of the bruised stalks into two inch pieces. Your end product should look like this:

For making salads
Slice each cleaned lemongrass stalk crosswise as thinly as possible. Separate into rings. Taste the outermost rings and discard if too tough or fibrous.
For making marinades
Cut each cleaned lemongrass stalk into one inch pieces. Smash in a stone mortar into a paste, or use a blender or food processor to pulverize into a thick paste (you may need to add a little water to the blender.)