LOL
Little did I know that when I was reading this thread last week that I would be in your same situation. My daughter broke the screen on her 5c over the weekend. We decided not to get a new phone because we were still paying the installments on this one (we have ATT next) and it otherwise worked fine. I also really didn't want to reward her for dropping her phone by getting a new phone (not implying anything with your daughter-mine did not pay for her phone and cannot pay for a new one). She had to pay for the repair.
I debated going to a cheaper screen repair place, but just went to the Apple Store. I'm glad I did. Because my daughter had the latest operating system update, we were told they could not only repair the screen. They gave her a new 5c phone for the cost of the screen repair $129 plus tax. That means a fresh battery, etc. She is also going to pay for a better case.
To get to your apple care question. We have not always ordered it on the phones that we have, but have always used it when we have it. iPhones seems to work really well the first year, but have issues going into year two, especially with the batteries. Since we are not upgrading as often as we used to, we are keeping our phones longer. I have also read that the iPhone 6 is being dropped more often because it is larger than the previous phones. I think it is a good idea.