Best way to learn is to get on and ride...that said...
The right shifter controls the rear. The "smaller" lever as you call it shifts to a smaller gear. The larger lever (brake lever) shifts to a larger gear. It the same way on the left side but you will be controlling the front.
You probably already know this but smaller on the back means that you pedal harder, smaller on the front means you pedal easier.
Have fun.
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Fredo, you're my older brother and I love you...but don't ever take sides, with anyone, against the family again...ever.