If you are considering a home treatment wax for the underarm area, you need to be quite skilled in waxing or the results may be somewhat disappointing (as well as painful).
Shaving is another option, but when you shave you are actually cutting off the hairs where they protrude from the skin and so they grow back stronger and thicker than before. This means you have to shave more often.
The easiest way may be to use a hair removal cream, of which there are many on the market, usually at a competitive price. You could also try treating the skin with a 'hair retardant' product after using the cream, to delay the hair from growing back.