Picture
http://sliderulemuseum.com/Ephemera/Apollo13MissionControl1.jpg
NASA learned a few important things from Apollo 13 that made the future expeditions more safe and successful.  First of all, they learned that no matter what happened, it was crucial to stay calm, or things would get way worse. 

Also, they learned that they needed to be more careful about what they put into the spacecraft.  The oxygen tank number 2 was installed in Apollo 10 at first, and was then removed for modification.  But while it was being removed, the tank was dropped 2 inches.  Though that may not seem like a lot for a durable spacecraft, the internal fill line was damaged slightly, and unnoticed during a later inspection.  The same tank was installed in Apollo 13, so in other words, a faulty tank was used instead of a new one where the chance of it being damaged are far, far lower than with a used one.  If NASA had simply replaced the tank, Apollo 13 might have been successful.  This proves my theory that only new and unused components should be installed in a spacecraft. 

Another lesson learned is creativity wound together with teamwork.  The crew in Houston worked together to immediately take action and use everything on the craft to the best of its ability, which saved the mission.  The command Module, where the astronauts live while in space, had lost power due to the explosion, so the astronauts were instructed to move to the Lunar Module, which was only meant to house two astronauts for two hours, not three astronauts for more than three days.  But without the move, the astronauts would not have been able to survive.  In the Lunar Module, there were not enough carbon dioxide scrubbers to clean the air in the craft for the needed amount of time, and NASA knew that.  But they quickly devised a tool that would successfully clean the cabin air with only the few materials available to the astronauts in the spacecraft.  Houston talked the astronauts through the process of constructing the makeshift filter.  It worked, and because of that last-moment innovation, the lives of Lovell, Swigert, and Haise were saved.