I strongly suggest you read at least the normal and true endings. The new version by Shiravune also adds a prologue to the story that was not part of MG's release, which is a bonus. The game is way too short if you focus on the main plot, and it will have many consequences in the second game.
If you don't want to go through it again, you should at least know what happens close to the end of the game (Stella's character arc, the truth about the second case - who was the killer, their reasons and how this plot point ended), as well as the whole Kuchiki family situation.
You don't need to play the bad endings, as they are more relevant to the first case, but in order to get to the true ending, you need to go through, I believe, at least one credits roll.
The second game does mention some things that happened in the first, but since you're playing two games already, another shorter one won't be an issue.