What is the oxidation state of S in ##S_2O_3^-2##?

2+
You should make sure to get this down solid, its important.
Oxygen is always 2-, and hydrogen is mostly 1- (there are some cases like water where it is 1+).
You add all the charges of the atoms to get the overall charge of the molecule.
So, ##S_2O_3^(-2) = 2S*(charg e) + 3O*(2-) = -2##
##2(charg e) + -6 = -2##
##(charg e) = -4/2 = -2##