Split your damage types to what your team needs. Yes AP Yi and AP Teemo are strong and fun, but if your team is already full of magic damage, then you might be better off going with an ADC/AD bruiser build. Even if you're traditionally a caster, you might add more going AD. I've seen good AD Malz/Annies win games.
The passive gold generation accounts for a lot of income. Waiting on 100 gold is worth it in a lot of scenarios if your team isn't in any danger.
Slightly past the first set of steps is where the shop locks you out. Keep in mind when it's advantageous to stay in base so that you can buy again later without dying.
Make the best of what your team has. If you have all one type of damage, try to coordinate an itemization with your team that grants a lot of reduction for that kind of damage (like Abyssal, Malady, and Cleaver.)
Ask for help. Chances are you're going to eventually get a champion that you don't know how to play. Ask your teammates what items are good, what to max, what runes/masteries to take, etc.
Health relics respawn every 40 seconds. Treat them like dragon spawns if you can.
Deny the other team health relics. If you're about to be in a situation where the enemy team will have control over the relics then take them for yourself, even if you don't need the health.
Give the relics to the person that benefits most from them. Having less health isn't what always dictates who needs the relic. A healer might benefit more from the mana, giving everyone more health in the long run. A melee champion is more likely to take damage and need the health than a ranged champion.
Let people with lifesteal/spellvamp attack minions to regain health.
The small collapsed holes at the edges of the map near the first sets of turrets block sight. You can use them for surprise skillshots, especially when the first tower is down.
The roots growing inside the base block vision. It's difficult to do so, but they can be used to LOS in a pinch.
Know what items do what kind of damage. If you're trying to diversify your damage, keep in mind that the vast majority of on-hit items deal magic damage.
Attacking from bushes grants sight. If your teammates are trying to stay concealed, don't give the enemy sight by attacking when it's not necessary to do so. Move out of the bush first.
Regen/GP5 items are good if you're forced to turtle. Philosopher's Stone is exceptionally good in this situation.
Stealth abilities can be used to suicide without giving the other team the kill. Waiting for the champion in the brush near the tower is an effective way to deny them the execution.
I'll post more if I think of some.