How the game behaves, in plain words.
Minion waves
Minion waves leave each base every 30 seconds and meet in the lane. Most of your early gold comes from last-hitting them, landing the killing blow on a minion. Where the waves meet decides who is safe: the closer to your tower, the safer you are, and the further your opponent has to walk to reach you.
Freezing the wave
Freezing keeps the wave still just outside your tower. Last-hit only, and let the enemy wave stay a little bigger than yours so it pushes towards you exactly as fast as your minions die. Your opponent then has to walk up to farm, far from their tower and in reach of your jungler. Use it when you are ahead, or when your opponent can't afford to lose minions.
Playing as a pair
In bot lane the two of you trade together: the lane goes to the pair whose damage and crowd control land first. Step forward together and back off together, and back at the same time when you can, so neither of you is left alone two-against-one.
Minion aggro
If you hit an enemy champion with an attack or a spell while you are near their minions, those minions switch to attacking you for a few seconds. Trading next to a bigger enemy wave costs you extra health, so trade when the enemy wave is small or your own wave is bigger. Enemy towers do the same: hitting a champion under their tower makes the tower target you.
When to back
Back when it costs the least: right after your wave crashes into their tower, when you have the gold for an item or component that changes a fight, or before an objective your team is going to contest. Backing with a big wave about to hit your tower loses minions and tower plates.
Measured below: First trip to the shop, Gold in hand then, Shop visits before 14:00.
Power spikes
A power spike is a moment your champion gets much stronger: levels 2, 3 and 6 (your ultimate), and each finished item. Fight when you have just spiked and your opponent hasn't, and avoid fights just before theirs. The matchup section of your game review in the companion shows the levels your lane opponent usually gets kills at.
Objectives
Dragons, Void Grubs, the Rift Herald and Baron give your whole team gold or power. Push your wave, back and buy before they spawn so you arrive first with full health. A team that arrives late, with waves crashing into its towers, loses the objective and the plates.
Measured below: With 2+ teammates from 20:00.
Vision
Wards show you where enemies are; a control ward also reveals and disables enemy wards near it. Ward the river or the enemy jungle entrance before you push forward, and place a control ward where you will fight later. Your trinket recharges, so use it whenever it is ready.
Measured below: Wards before 14:00, Control wards before 14:00, Wards a minute.