To care for your lettuce plants, make sure that they don’t get too hot and keep the soil moist. Feed lettuce regularly (frequency depends on the type of fertilizer you use, so read the lable) with a liquid, high nitrogen fertilizer, like diluted fish emulsion or compost tea (though if you are using compost tea, make sure it's aerated, or there is some risk of spreading E. Coli bacteria).
How you harvest lettuce depends on what type you have planted. You can pick the leaves of leaf lettuce from the outside, letting the inside leaves continue to grow. You can also cut off the whole plant, about an inch above the soil and let it re-grow.
For lettuce that grows in “heads” you usually will pull up the whole head.