So I learned that early game, health lets you live longer than armor does.
That depends on the situation.
Health is more cost-effective on a gold-per-power bases than either Armor or MR, as at level 1 you only need ~500 HP to double your survival, with armor/MR you need ~130 on top of your bases, which is more costly.
However: this mostly assumes a burst-war aiming to 100%-->0% a target.
Armor and MR have much greater synergy with any healing, HP bonuses, or when facing opponents with life steal. Restoring 10 HP when you have 50% damage reduction ultimatly accomplishes the same thing as restoring 20 HP with 0% damage reduction, as your opponent needs to send 20 damage your way to counter it in either case. As such: HP/5, healing potions, health relics, life-steal, spell-vamp, healing, HP shields, and anything else that bolsters HP is directly improved with Armor/MR, while HP has no innate synergy with such mechanics.
As the lanning phase tends more towards a contest of endurance than burst, Armor/MR investments can compound their defensive bonuses many times over between trips to the fountain, where HP generally just offers it's bonus once between trips back to the base.
Armor also helps a lot vs. hostile life-steal, as doubling defense through HP will have no impact on the HP restored per AA from a vampiric champion, where halving their damage output with armor/MR will also halve the healing they revive per AA/ability for double cost-efficiency.
Another thing to consider is that while counter-building is certainly a thing in LoL, the advantage counter-building grants is usually pretty minimal compared to the synergy provided by completing core items for a champion.
It's nifty if both line up, but it's almost always optimal to go for items that resonate with the champion you're playing over items focused on giving your lane opponent a hard time, regardless of what Riot tried to tell you with LoL's tutorial recommending 
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