React contains both the very first Melty Blood story [with an updated battle system - this is really important] and a sort of sequel story told through the Arcade Routes. Act Cadenza IIRC shares most of the same story as React with some minor changes; and additional stories for new characters that were added later on. MBAACC is the sequel to both and IIRC the story, for the most part, ignores Act Cadenza and much reference is made to the first game.
Do not play the original Melty Blood.
Furthermore, Melty Blood is in itself a sequel to another franchise called Tsukihime, so if you want further backstory on some of the characters, you could look into that too.
All in all though, the story isn't very important in Melty Blood, so you can just jump into MBAACC if you want.