I'm going to disagree on the above interpretation of Nightwing: Year One, but I can definitely see how it can be see as him being a thug. Personally, I took it more as Jason's overeager bravado at trying to prove he deserved to be Robin and him falling back on what he's familiar with in a situation that went very off script (six months of Robin training wouldn't train out his street smart instincts either). It also shows Batman's less than likable traits as a mentor and father figure toward Jason which is important considering everything that happens with Jason and how his relationship with Bruce influences everything in his life. It's not nearly as bad as some of the other flashbacks of him, but it does show some of Jason's less likable traits as an early Robin. He was 13, not perfect.
Also, you really should read Batman Annual #12, it's got a pretty important Jason story in it about his school life and his training as Robin (and it along with NYO also show that Jason was every bit as good performance-wise as the other Robins despite constantly being bad mouthed in canon). There's also The New Teen Titans v1 #19-30, as it shows that Jason was indeed a Teen Titan for a while (and one of the few brotherly scenes between Jason and Dick is in there), and Teen Titans v3 #29 because that's some what he was up to in between the two mini-arcs of Under the Hood--and actually part of it relates back to The New Teen Titans arc.
Before reading Under the Hood, you really should read the Hush Arc (Batman #608-619). There isn't a whole lot of Jason in this arc until toward the end and some of it is kind of twisted and confusing, but it's kind of important in regards to Batman's feelings about Jason and Lost Days and Batman Annual #25, which are basically Jason's post-resurrection/pre-Red Hood backstory issues. It is also the first hint of Jason returning to DC that you get. As for Countdown, I would also say read The Search for Ray Palmer side issues, particularly Gotham by Gaslight.
The issues in Batman and Robin are important, depending on what canon point you want to take him from. The first arc (end of #3 through #6) is in many ways one of the worst portrayals of Jason ever (it's right up there with Battle for the Cowl, yet at least that can be explained in some twisted way). But it was written by Morrison, so just expect BAD going in (only thing he got right was that Jason isn't against sidekicks at all--despite some people arguing he hates them). Jason's relationship with Scarlet is important to the next arc he's in (#23-25) and despite having to go along with SOME of the things from Morrison's arc for continuity's sake, those issues were written by Winick, which is guaranteed amazing Jason characterization. So even if you don't take him from that point, read it just for how well Winick always handles Jason as Red Hood. To be honest, some of my most favorite Red Hood quotes are from those issues. Unfortunately those issues are left feeling open-ended and unfinished, but Flashpoint and then Reboot happened.
DA
Also, you really should read Batman Annual #12, it's got a pretty important Jason story in it about his school life and his training as Robin (and it along with NYO also show that Jason was every bit as good performance-wise as the other Robins despite constantly being bad mouthed in canon). There's also The New Teen Titans v1 #19-30, as it shows that Jason was indeed a Teen Titan for a while (and one of the few brotherly scenes between Jason and Dick is in there), and Teen Titans v3 #29 because that's some what he was up to in between the two mini-arcs of Under the Hood--and actually part of it relates back to The New Teen Titans arc.
Before reading Under the Hood, you really should read the Hush Arc (Batman #608-619). There isn't a whole lot of Jason in this arc until toward the end and some of it is kind of twisted and confusing, but it's kind of important in regards to Batman's feelings about Jason and Lost Days and Batman Annual #25, which are basically Jason's post-resurrection/pre-Red Hood backstory issues. It is also the first hint of Jason returning to DC that you get. As for Countdown, I would also say read The Search for Ray Palmer side issues, particularly Gotham by Gaslight.
The issues in Batman and Robin are important, depending on what canon point you want to take him from. The first arc (end of #3 through #6) is in many ways one of the worst portrayals of Jason ever (it's right up there with Battle for the Cowl, yet at least that can be explained in some twisted way). But it was written by Morrison, so just expect BAD going in (only thing he got right was that Jason isn't against sidekicks at all--despite some people arguing he hates them). Jason's relationship with Scarlet is important to the next arc he's in (#23-25) and despite having to go along with SOME of the things from Morrison's arc for continuity's sake, those issues were written by Winick, which is guaranteed amazing Jason characterization. So even if you don't take him from that point, read it just for how well Winick always handles Jason as Red Hood. To be honest, some of my most favorite Red Hood quotes are from those issues. Unfortunately those issues are left feeling open-ended and unfinished, but Flashpoint and then Reboot happened.