Your living lettuce came home with its roots still attached — that's what makes it last so much longer than anything from a grocery store. With a little care, you can keep it fresh and crisp for a week or more.
Place the whole plant upright in a container or jar with about half an inch of water covering the roots. You can loosely drape a plastic bag over the leaves to hold in humidity. Keep it in the fridge and change the water every two days. Peel off outer leaves as you need them — the plant will stay alive and crisp for 7–10 days this way.
If you plan to use it within a few days, you can keep the plant on your counter in a glass of water with indirect light. It may even put out a little new growth. Just keep the roots submerged and change the water daily.
Don't separate the leaves from the root ball until you're ready to eat them — attached leaves stay fresher much longer. If the outer leaves start to wilt, peel them off and the inner leaves will still be perfect. Rinse leaves gently under cool water right before eating.