The intro. Ann Druyan's tone, the passion with which she speaks, shows that she and Sagan were indeed Happily Married, and that she believes in his cause just as much as he did.
The new series just started and already it had a heartwarming moment of epic proportions: Neil deGrasse Tyson recounts his first meeting with Carl Sagan, and says that before meeting Sagan, he already knew he wanted to be a scientist, but after meeting Carl, he knew what type of person he wanted to be. Manly Tears were shed.
Here's what happened: Tyson shows the audience Carl Sagan's personal calendar from 1975, showing the date when he met with Tyson. Sagan personally picked up Tyson from the bus station in Ithaca and showed him around Cornell University, even giving him an autographed copy of The Cosmic Connection. After the day was done, Sagan took him back to the bus station. Concerned that the snow was falling harder, Sagan gave Tyson his home phone number and told him to call if the bus couldn't make it, offering Tyson to spend the night with his family. The idea that the world's most famous scientist at the time would devote that much attention to "just a seventeen-year-old kid from The Bronx" shows what type of person Carl Sagan was, and why Neil deGrasse Tyson wants to pay it forward by making science popular again.
Here's an Imgur .gif gallery for the opening for those who want to view it.
Episode 3 of the reboot portrays the friendship between Isaac Newton and Edmund Halley, without whom Newton's Principia would never have been published. This event is then linked directly to the exploration of space in the 20th century, which was made possible by Newton's ideas. Tyson then notes that Halley has been in his friend's shadow ever since despite his own considerable scientific work, clearly hoping that he's doing his part to restore the reputation of this forgotten giant.
The depiction of William and John Herschel in Episode 4.
After the finale aired, Anne Druyan tweeted this: "The Ship is now yours."
Tyson talking about Voyager, traveling alone past the heliosphere. A lone relic of humanity... that will last at least a billion years.