Great idea for a thread, Xela!
+1 to Shawn's comment on gears. They are there for a reason. Learn not only to use them, but also how NOT to use them. Trying to shift between front rings under full power, for example, will not work well at best, and will leave you stranded with your chain stuck between the rings at worst.
Anticipate the need for a lower gear and make your shift before you're straining to climb the hill. Later, learn to shift while climbing by spinning the cranks with very little 'torque' just long enough to catch the next ring, but before you lose all your momentum and fall over
Don't forget that your derailleurs have VERY little 'power' to work with in making a shift. Your push of a thumb, twist of a wrist, etc. is all the energy they have available. It isn't much next to your legs.
Clean your chain regularly and keep it well-lubed. Too much lube is as bad as too little here. A little bit goes a long way. A properly-lubed chain reduces mechanical drag and lets you put more of your pedal-stroke toward moving yourself and the bike forward rather than working to circulate the chain. It goes without saying that the chain will last longer when it's properly lubed. If you notice squeaking out of your drivetrain while riding, it's probably been too long. Stop. Oil your chain.
Adjust. Your. Seat. Height. Please. Doing so will literally make you a stronger rider. When your seat is too low, you aren't giving the muscles in your legs the ability to fully engage the cranks to move you forward. A too-low seat also puts inappropriate stress on your knees. Over time, this can damage them badly. It takes time to get seat height dialed in perfectly, but you should see an immediate gain in power and comfort from even gross adjustments. That said, too high is just as bad, if for different reasons, as is too low.
Never leave home without at least a patch kit, pump and preferably also a spare tube. If you run 'slime' or one of its analogs, forget the patch kit and just carry a spare tube - some road debris will take out your slimed tube like it's not even there. It's great to not have to walk when this happens, as it did to me just this weekend - I never saw what it was, but hit something that left a 1/4" gash through both the outside and inside of my tube. Slime or no, that one's done.
Even if you're not a 'slime' person, this kind of situation is why it's wise to still carry a spare tube in addition to the patch kit. You (hopefully) won't need the spare tube often either way, but when you do, it makes up for all the miles you've carried it. Your experience may vary. Don't forget your tire levers
I count them as part of the patch kit.