1. Are you sure that you heard your grandfather right? That's not a stock 9-speed cassette. Shimano road cassettes usually end at 27 teeth and Shimano mountain cassettes end with 32 or 34 teeth.

2. To change your cassette you'll need a cassette lock ring tool, a chain whip, and a big crescent wrench.

3. You can almost never change just one part on a bicycle because everything works together. In this case you should probably get a new slightly longer chain. The rear derailleur that you have will probably work with a 30 tooth big cog but will probably need to be replaced with a mountain bike rear derailleur with a 32 or 34 tooth big cog.

4. Your bike shop lady was probably suggesting that, insted of messing with your cassette, you replace the 30 tooth front chainring with a smaller one. That would almost certainly be cheaper, would still give you the easier hill climb gear, and would eliminate the big steps between gears for the flatter majority of your riding. That's what I would do too.