Who Hits Beginning the Next Inning

[officialtony]

OK,
I can't imagine you don't already know the answer to this, but here you are.
Presuming less than two outs -
Follow the sequence of events:
Batter swings at a pitch for strike three - he is out;
Catcher throws baseball to base to put out base stealing runner;
Base stealing runner is out.

If these are outs number one and two, then the next batter in the batting order following the batter who struck out will appear at the plate.

If this is out number three, the batter who immediately follows the batter in the batting order who struck out will come to the plate in their half of the next inning.