Gatkins - you are correct on the basic concept: the gear ratio is the number of teeth on the front chainring divided by the number of teeth on the cog in use in the rear. The lower that ratio, the more times you will have to turn the pedals to move a given distance or maintain a given speed. When climbing hills, that translates to easier

How much easier you need is related to your fitness, the weight of you plus the bike and anything you are carrying, and how steep a hill you plan on climbing. Most touring bikes have a very low gear (under 1:1) because even if the rider is a skinny balink, the bike and the luggage is going to be pretty heavy and the fitness level of the rider may be really pretty low. Most racing bikes don't have such low gears because the rider is supposed to be skinny balink, have high cycling fitness, and the bike is lightweight and carrying nearly zero weight other than the rider. Yet, both are used to climb very steep hills, just at very different speeds!

Someone else pointed out the another consideration: how high a cadence and how fast you want to go on level ground and downhill. On a multi gear bike, the rear cassette and derailleur limit how wide a range you can cover in gear ratios and the largest front chainring determines if you will have to pedal like a Tasmanian devil to maintain your desired speed when not climbing.