The most obvious choice is to make a bowl of aromatic Thai soup. To make an easy Tom Ka Gai (coconut chicken soup), heat a little olive oil in the bottom of a pot. Toss in a couple of tablespoons of red curry paste, a couple of tablespoons of grated fresh ginger and a minced clove of garlic and sauté until fragrant. Add a cubed chicken breast, and cook the meat through. Add four cups stock (either chicken or vegetable), a couple of tablespoons soy sauce or fish sauce, a spoonful of brown sugar, three cans of coconut milk and a bruised stalk of lemongrass. Cook until thickened, remove the lemongrass, squeeze in the juice of a lime and garnish with fresh cilantro.
Lemongrass is also appropriate in other soups and savory dishes throughout Southeast Asia (but most notably Vietnam and Thailand). One of the easiest ways to use it is in a marinade for Vietnamese Barbecue Beef or Chicken, including minced lemongrass in a mixture of fish sauce and aromatics (such as onion and garlic), curry powder, and salt and pepper. Serve it with rice noodles and lettuce leaves, bean sprouts and matchsticks of veggies.
Lemongrass is a must in most varieties of thai-style curry paste. This is an instance where you do eat the whole herb, as it is pounded down using a mortar and pestle. These curry pastes can be bought cheaply in cans at Asian markets, or you can make your own. Experiment with different varieties (named mostly for the color of the paste) to find your favorite. Yellow curry paste is the mildest (as it relies heavily on turmeric) and red has the most assertive notes (it is often paired with beef and spicy chilies).
Don't forget to consider lemongrass for fish dishes, even if they are not of Asian origin. A few stalks can be laid under fish in a steaming basket, and will infuse the lemony flavor during the cooking process. If you are cooking a whole fish, bruise a few lemongrass stalks and use them to stuff the cavity.
You know the rule about how you can have lemon in your tea, or you can have cream in your tea, but you can't have both? This is because lemon tends to curdle dairy (a good thing to know if you need a substitute for buttermilk in a pinch). But lemongrass won't. Consider steeping this herb in cream for savory cream sauces. It can even be used in ice cream. Just take your favorite vanilla ice cream recipe, and gently heat the dairy, then allow the lemongrass to steep in it for about half an hour. Strain it out, let cool, and proceed as required by the recipe. If you want to add inclusions, consider coconut, or even slivers of crystalized ginger.
This herb is also known in many parts of the world as fevergrass, where it is often drunk as a tea for medicinal purposes (this should be done with caution, especially if you take prescription medications, as there can be interactions). A variation on this is to steep the lemongrass, strain it out, add honey, ice the liquid and serve it as lemongrass lemonade.
Once you become familiar with the taste and texture of lemongrass, you can experiment with ways to use it to add lemony flavor to everything from baked goods to flavored vinegars. It is another way to let your herb garden take you on a trip around the world.
Note: You only cook the stems of lemongrass, where the grass wraps around itself. The top blade, while not useful for cooking, are still fragrant, and can be dried for use in potpourris.
Note: Lemongrass has a texture reminiscent of a fine nail file. If you run your finger the wrong way across the edge of the grass, you can cut yourself. Some people slice the stems very finely and eat them in the finished dish. I find that even in small size, lemongrass grits against your teeth, like when you are having them polished at the dentist. You can get the flavor without the grit by bruising the stalks (smash them with the side of the blade of a heavy knife) and adding to a dish whole, allowing the herb to simmer, and then removing just before serving, as you would a bay leaf.