Based on the other thread, you have a derailleur drivetrain, with three sprockets (chainrings) in front, and a cluster of seven sprockets located on the rear wheel hub.
Derailleur gearing is conceptually pretty simple. You have two derailleur mechanisms controller by the shifters; the front is a cage above the chainrings which the chain passes through, the rear is a more complex mechanism below the rear cluster which includes a two-wheel tensioning pulley. To shift gears, they both work by moving to the left or right, to cause the chain to
derail from the current sprocket and hop to the next.
What determines the "difficulty" of a gear is the ratio in tooth count between the front chainring and the rear sprocket. A 52-tooth chainring on a 13-tooth sprocket is a 4:1 ratio, typically considered fairly high. Whereas a 28-tooth chainring on a 28-tooth sprocket is a 1:1 ratio, a pretty dang low gearing ratio that some bikes don't even go down to, and which would typically only come into play while climbing a steep hill. The bigger the ratio, the farther the bike will move with each pedal revolution, and the higher the gear. To make your gearing higher, shift to a bigger chainring in front or a smaller cog in back; to make your gearing lower, shift to a smaller chainring in front or a bigger cog in back.
Usually the front shifting isn't quite as smooth as the rear shifting, especially when moving from a smaller chainring to a bigger one. So people frequently use front shifts to select a "range", such as shifting to the small chainring to set their bike into the low gears when a hill is approaching, but do most of their shifts on the rear cluster.
When you use a small chainring with a small cog, or a big chainring with a big cog, the chain is running fairly diagonally, a condition called "cross chaining." This can make noise and increase wear on the drivetrain. To solve this, you can shift in front and rear at the same time to find a combo that will have a similar ratio without the cross chaining; for example, large chainring and large rear sprocket will often have about the same ratio as one of your gears in the middle chainring and a middle rear sprocket.
Big-big is frequently even more problematic than small-small. Because the chain is being stretched around two large sprockets, the rear derailleur's tensioning mechanism can sometimes be getting pulled to its limits. The resulting "binding" can introduce noticeable friction and a whole lot of noise. In very extreme cases, where people have set their bikes up with chains that are super short, you could even rip off the rear derailleur by shifting into the big-big combination.
As far as what gear you should be in, it depends. Typically people will aim for a certain pedaling cadence; typically appropriate cadence varies by intensity, where higher-intensity cycling demands higher cadence. So, if you're trying to maintain constant intensity, you'd shift down when you get to a hill because there's more resistance.
There are other complicating factors; for instance, pedaling out of the saddle effectively lowers your body's gearing, making it appropriate to use higher gears on the bike and pedal at lower cadence to maintain similar intensity.
People tend to figure out pretty good gearing strategies naturally as they get more experienced with cycling. What feels awesome is usually not that far off from what's appropriate.
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Also, never refer to what gear you're in with a "number" like 1/2/3/etc. Tooth counts, words like "high" and "low", and words like "big" and "small" are all intelligible. Describing your gears in
gear inches or distance of development is also intelligible, but only to people who know what they're talking about. If you use 1/2/3/etc notation,
we're going to yell at you.
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If the rear sprocket seized against back-tension on the chain rather than freely spinning backwards, you'd wreck the rear derailleur: the chainring would be yanking chain away from the tensioner, but the rear sprocket wouldn't be giving the tensioner new slack to work with, so the pulley would basically get ripped forward and right off the bicycle. So, derailleur gearing is incompatible with fixed gears and coaster brakes.
A freewheel that allows backpedaling is also the simplest design for a freewheeling mechanism. Unless there's some extra feature that works against it, such as a coaster brake, a freewheel that allows the rear wheel to turn forward without pushing on the chain will also allow the chain to turn backwards without affecting the rear wheel.