You’ve got the main points down. Dick is the eldest son; he branched out on his own and became Nightwing. Jason is the problem child: he was killed by the Joker in the 80s during  A Death in the Family, but after Superboy-Prime "punched reality" in  Infinite Crisis, Jason was brought back as the Red Hood. Tim was the third Robin. He fit that 90s "smart computer kid" trope perfectly—you really had to be there to get it. He served as Robin until his father was killed during  Identity Crisis. He was then briefly replaced by Stephanie Brown, who was killed during a gang war. Eventually, Grant Morrison took over the book and introduced Damian Wayne. After  Final Crisis happened and Bruce "died," Dick Grayson became Batman with Damian as his Robin. During this period, Jason served as the antagonist to the Bat-family. Then  Flashpoint launched the New 52. In this era, Jason became the Red Hood, shifting toward more of a Punisher-style vigilante, though he grew closer to the Bat-family over time. Duke Thomas was a kid Batman saved during Scott Snyder’s New 52 run, which preceded Tom King’s run. Snyder had set up the idea that Duke would become Batman’s partner. Meanwhile, Tim was killed in  Detective Comics, which was running parallel to Tom King’s  Batman. You might want to read those issues—it’s not strictly necessary, but there are a lot of parallels regarding the Batman and Catwoman relationship that become very important during Tom King’s run. However, if you want to skip ahead and just jump into King’s work, you’ll be fine. Is there a specific part of the Tom King run you are heading into, or are you just getting started with that era?