Question:  How in the heck do I solve equations using molarity and molality?

Answer:
The answer:  It depends on the equation.

To use either of them effectively, it's important that you know the difference between the two and when you should use each of them.

Molarity is equal to the moles of solute per liter of solution.  Molarity is by far the most commonly-used unit of concentration, and is used to calcuate pH, in equilibrium expressions, and to denote the concentrations of just about every solution you'd ever
use in the real world.  Real chemists can usually get by without using anything else - when I worked as a chemist, I had to look up the definition if I used any other unit of concentration because it was so uncommon.  (Normality is a unit closely based on molarity, and is equal to the number of equivalents of stuff per liter of solution.  This is a handy unit for applications where the number of each type of ion is important, rather than just the overall concentration.  Chemists use this unit a lot, too, but I classify it as being practically the same thing as molarity, since they're both so similar.)

Molality is equal to the moles of solute per kilogram of solvent.  Even though one kilogram of water has a volume of one liter of water, the molality of an aqueous solution isn't the same as the molarity.  The reason for this makes sense if you use an example:  If you put 58 grams (one mole) of NaCl in a liter of water, the final volume of your solution will be something greater than 1 liter.  As a result, the molarity will be less than 1M, while the molality will be exactly 1 m.  It's usually not a big difference,
but enough to screw up your answers.

About the only time I've ever seen molality really used in when doing calculations with colligative properties.  For some reason or another, these calculations require molality, so you've got to remember to use it here.  Molality is probably used for something else somewhere, but I can't offhand think of what it is.