1. Remember to Treat Yourself
Working at Seattle Grace Mercy Death (now grey Sloan memorial, in honor of two dead doctors), the doctors of Grey's Anatomy never know when they will die. Thus, they make a point to enjoy whatever time they have. First thing they did after leaving the hospital was head to the bar for a relaxing drink, so I'd say they're living it to the fullest.
Very few of the characters on the show were raised in happy homes, and when they all work 80+ hours a week, they soon realize they have to connect with each other. Thus, they begin sharing their homes, their hearts, and their deepest secrets with one another. While not blood, they know the love they have for each other makes them a family.
3. There's more than one right way to do a jobAs an audience, we quickly notice the different bedside manners of the surgeons. Cristina, for instance, is distant and objective while George is warm and empathetic. Most of the other doctors fall somewhere in-between these two extremes, and all of them work hard to try and create the best outcome for their patients. Rather than value one over the other, they show the good and bad from both approaches.
4. If you love someone, let them knowAgain, part of the reason we learned this is because of the large quantities of death and destruction, but it resonates all the same. Far too often we take the people in our lives for granted, not telling them how important they are to us until it is too late. We assume that we'll have more time, never thinking we won't get the chance to tell them how we feel. Or, more often, we fear the timing or possible response. While the show isn't exactly realistic, it does touch on something universal.
5. Death might be inevitable, but it's still devastating.