I am not sure what you are really asking, but let me try to answer anyway. I understand that you refer to the Rabin/Scott construction discussed in

https://q2a.cs.uni-kl.de/1137/sheet-9-exercise-rabin-scott-determinisation-not-accepted?show=1138#a1138 and you wonder how the initial states and accepting states are defined.

The initial state (it is only one in a deterministic automaton) is the set of initial states in the original automaton. The accepting states are those superstates which contain at least one accepting state of the original automaton.

In the mentioned example, the original automaton had initial states {s2,s3} so that is our new initial state (which was arbitrarily abbreviated as state Q0 of our deterministic automaton). The accepting states of the original automaton were {s0,s1}, and therefore states Q0:{s0;s2}, Q2:{s1}, Q4:{s1;s3} and Q5:{s0;s1;s2;s3} are the accepting states.

You further question is why we call the states s0,s1,s2,s3 instead of just 0,1,2,3 as in the text description and why they are called Q0,...,Q5 in the deterministic automaton. That is a matter of taste. The states could be given any names. In the images we add "s" or "Q" to avoid confusion in the discussions when we otherwise would just talk about state 2.