Usually, when we are balancing equations, we need to count the individual atoms which will certainly work in this case as well but when there are polyatomic ions that do not change between the reactants and the products, we can simplify the process.
Ca(NO3)2+KOH --> Ca(OH)2+KNO3
When we look at our equation, we see that we have 1 Ca^2+, 2 NO3-, 1 K+, and 1 OH- on the reactant side and 1 Ca^2+, 2 OH-, 1 K+ and 1 NO3- on the product side.
Placing a 2 in front of KNO3 will balance the nitrates
Ca(NO3)2+KOH --> Ca(OH)2 + 2KNO3
But now we need a 2 in front of KOH to balance the K+ ions
Ca(NO3)2+ 2KOH --> Ca(OH)2+ 2KNO3
As a result, it also gives us 2 OH- ions, so we have two of those on both sides. Now we can go back through and check and find that we have a balanced chemical equation.