Gears sizes and force
Q. If you want to minimize the force you exert on the pedals what sizes gears should you have at the crank sprocket (at the petals) and the freewheel sprocket (at the wheel)?
A. At the crank sprocket (at the petals), you want to be on the smallest gear. On the smallest gear you are moving the smallest amount of chain. Since the amount of chain you move changes how much the wheel goes around, moving less chain means moving the wheel less, which means the bike is doing less work on the road (less distance), so you must be doing less work on the bike. Since you're feet are going the same distance around the pedals, you must not be pushing as hard (less force).
At the freewheel sprocket, it's the opposite; you'll have to apply the smallest force on the largest gear. There are two ways to think about this. First, on a large gear you are further from the center of the wheel when you apply the force. This means you have a longer level arm, so it takes less force to spin the wheel. If it's easier to spin the wheel then it's easier to move the chain and thus easier to pedal. The other way to approach this is that on the largest gear it takes more chain to spin the wheel. That means when you go once around on the pedals and move some amount of chain this amount of chain only makes the wheel go around once or twice (as opposed to five or six times on a smaller gear). Moving the wheel less means the bike is again doing less work on the road (less distance) and thus you must be doing less work on the pedals (less force).
When the pedals are easiest to push (least force) you are doing less work each time you push the pedals once around, but you are also moving less. To go as far as on other gears you're going to have to turn the pedals around more times (a greater distance). Regardless, you have to do the same amount of work to get where you're going!