Why isn't the Beast Titan at the start of season 3? Season 2 ended showing the characters, some titans and then on top of a far away wall the Beast Titan with the titan shifter outside him looking out, but now on season 3 there was only a brief mention of him.

Is the story continuing right after the previous episode or is there any time gap between them?

Will it go back to him any time soon in the next episodes?
Expect the Beast Titan to appear very soon, this season of Attack On Titan will be very packed with its 24 episodes. The story is continuing after season 2's last episode, but there was a little time skip for both parties to kind of regen and recouperate from last episodes antics. Best way to enjoy this season and Attack On Titan is to stay far away from spoilers and let everything naturally unfold the way it should do.

Hope that helped.