Yes, there is rebirth in the suttas, but it is not rebirth of a specific person or a self or a transmigrating soul. It is the "rebirth" or continuation of suffering. Suffering has been going on for a very very long time, without a clear beginning point (SN 15.3). Suffering that continues whenever and wherever there is craving (SN 44.9). And craving appears whenever and wherever there is ignorance. Suffering does not end with physical death (which is itself suffering). Suffering is only ended by Nibbana. Nibbana is the only "place" where there is no suffering.
In my opinion, whenever the suttas speak of the recollection of past lives, it is not the recollection of the past lives of a specific person, but rather the recollection of the past lives of suffering. And whenever the suttas speak of "rebirth" or birth, it is also the "rebirth" or continuation of suffering.
Kamma and its consequences also exist. It is simply the continuity of the cause-effect chain of dependent processes.