I can remember cups and gallons thusly : "A quart is a quarter of a gallon and also a quartet of cups". Thus there are 16 cups in a gallon.
Similarly, tbsp:cup as cup:gallon, or "the gallon's cup is the cup's tablespoon". So there are 16 tbsp in a cup.
In a system of powers of two this all sort of makes sense. Powers of two makes sense because you can combine two of one measure to make the next measure. 16 is a nice place to put the major divisions. If you like, 2^8 tablespoons is a gallon. A gallon is a byte, a cup is a hex digit, and a tablespoon is a bit.
Two tablespoons are an ounce. If you like you can remember this in a weird way : first of all "a pint's a pound the world round". There are ounces of volume and ounces of weight, but they are the same for water. Since a pound is the fundamental unit of weight, 16 ounces make a pound, since 16 is our primary division and the ounce:pound as cup:gallon. Thus since a pint is double a cup, an ounce must be two tablespoons.
Three teaspoons are a tablespoon. This one is trouble.