calculate the molarities of each of the ions in a solution prepared by diluting 78mL of 4.5M Ca(ClO3)2 (aq) to 156mL with water. I don't understand how to answer this question.

First determine the concn of the molecule. That is done one of two ways;

1)
4.5M x (78/156) = ? or
2) mL1 x M1 = mL2 x M2.
You should get the same answer the same way.
Then you know there is 1 Ca ion is 1 molecule of Ca(ClO3)2 so M Ca ion is the same as M of the Ca(ClO3)2.
You know there are two chlorate ions (ClO3^-) in one molecule of Ca(ClO3)2; therefore, concn of chlorate ion will be twice that of the molecule.