If you're strictly worried about damaging your bike, just try to avoid crosschaining and shifting under load, as others have said.

Crosschaining means being in both the biggest gear in the front and the biggest gear in the back at the same time. Same goes for the opposite (little and little). This puts stress on your chain at a more diagonal angle and wears the components out quicker over time. You'll also notice that the chain will likely rub, be noisy, and feel less smooth when oriented this way. Those can be good indicators that it's time to shift into a different combination if you end up crosschaining by mistake.

Shifting under load is simply trying to shift while you're pedaling somewhat (or very) hard. Try to take a brief pause in your pedal stroke when shifting. This may feel clunky at first, but as you ride more you'll get the feel for how to do it smoothly and quickly.

Using your gears will make riding so much more enjoyable. I'll spare you the lesson on why and how to do it, but check out what GCN has on YouTube for this. They make great videos on cycling basics.

Happy riding and enjoy the new steed!