Here's Sheldon Brown's gear calculator, to compare various setups. That whole site has a wealth of technical information. RIP Sheldon.
It's really not that complicated. You want to know for every revolution of your legs, how far will your rear (drive) wheel travel? Let's say you're in a gear where your rear cog has the same number of teeth as your front one. Maybe a 30x30. (That's a very low gear, stock road bikes don't normally go that low, but mountain bikes do - anyway just for the sake of discussion.) For every revolution of your crank, your rear wheel will also make one revolution.*
Okay, now shift your imaginary derailleurs into a gear where your rear cog has exactly half the teeth of your front one. Say it's a 42x21. Now, every time you move your legs and crank one revolution, your rear wheel will make TWO revolutions. So you're going twice as far for every revolution of your legs (faster given a fixed cadence), but you also have less mechanical advantage (you have to exert more force on the pedals to go the same speed).
Bigger cog in front = more revolutions of the rear wheel per crank revolution ("more speed"); bigger cog in back = fewer revolutions of the rear wheel per crank revolution ("more power").
HTH to get you started. There's more to think about, but that's a beginning.
*"Gear inches" is based on the diameter of the wheel, not the circumference. I don't know why.



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