I thought that Batman #1 picks up after the one-shot Batman Rebirth issue. Fortunately, it isn't! This one is a continuation of Snyder and Capullo's epic New52 Batman run. Do you need to read the New52 Batman series? It is not necessary, but I highly recommend that you start there. The New52 Batman series is one of the few things that came out great from that era.
Unlike Detective Comics, Batman comics is obviously a Bat-centric series, and here we see what the Caped Crusader can do. If you look at the issue cover, that pretty much summarizes the plot. There's Duke who has been introduced to us in the Rebirth one-shot (plus appearances during Snyder's run), Commissioner Gordon, an airplane with its tail on fire. Alfred is also in the issue. Then there's this couple in blue and gray, with a pentagon insignia ala-Superman, but with an old English letter of "G". They seriously have to work that letter because at first look, it looks like a "B".
So Batman issue one doesn't really introduce these two characters, save for the last few pages, where some of their powers are revealed. They are heroes by the way. This issue is an entertaining read from start to finish. Finch's art style is good as ever and Tom King is a competent writer like Snyder. I just miss the latter's narrative style.
We also see Batman's new costume after a hiatus in the wake of the events in Endgame and after Commissioner Gordon took up the mantle for a brief period. I like the new costume which kinda respects the old and the new.
Is this a good start? Yes and so far I love what I am reading from DC Rebirth. Is this as good as Snyder's Batman issue one? I'm leaning towards Snyder's first issue. It was much more enjoyable. Is this better than Detective Comics #934? I cannot say, although I enjoyed DetCom more.
The Batman series is alive as always. It is promising, it is both fresh and familiar.