You're not going to strip your gears.

Big chainring at the front - high gear, harder to pedal but you go faster. Small chainring low gear, easier to pedal, good for hills.

Its the opposite at the back, the big sprocket is the easy one, the smallest one the hardest. So the easiest gear, for hill-climbing etc., is small ring at the front, big sprocket at the back.

As avidone has suggested, just back off the pedalling pressure a bit - but don't stop pedalling - for a smoother shift. Those are friction shifters, they aren't indexed, so you'll need to get used to the "feel" of how far you need to move the lever to get the chain to shift across a single cog. Just mess around experimenting for a day or two, you'll be surprised how quickly it becomes second nature.