Runes allow you to improve your character outside a regular game of League of Legends. There are four types, Marks, Seals, Glyphs and Quints.
Marks tend to boost offensive categories, Seals are defense-focused, Glyphs favor magic, while Quints are good all-rounders and are the best type of Rune (although are far rarer than the others, with only 3 allowed at maximum, compared to 9 for the rest).
You need to first buy Runes from the store, then you can activate them via your Profile page, and the Rune tab.
Quintessences are a type of rune, the rarest one. You can only begin to use one at Level 10. Then you unlock a second at Level 20, and the final third at Level 30. This is far fewer than the other runes that you are allowed.
To equip them, once you have reached Level 10 (they will not show up before then) and purchased one from the store, head to your Profile page, then the Rune tab. Left click on the one you want, and move it to the Quintessences section.
Making sure you have all the hotkeys that you need set up is vital for succeeding at League of Legends. The best way to get them organized is to play a Practice match. Press Escape, then go to your key bindings menu. From here you can assign hotkeys and practice with them to make sure they are practical in combat.
Influence Points are used to purchase many items from the Riot Store. While there's no easy way to earn them, they are accumulated as you play, regardless of whether you win or lose. Obviously the best way to earn Influence Points is to win, and in as quick a time as possible.
You will want to make sure you get the first win of the day bonus as often as you can, this will give you a significant lift. You could also buy an IP Boost using Riot Points. Bear in mind that playing PvP rather than against Bots will always give you more IP also.
To level up in League of Legends you should treat it like a MOBA, and not an MMORPG. What is meant by this is that you should primarily just play your favorite mode as much as possible if you want to level up quickly, rather than trying to grind your way through them.
You can invest in a XP Boost and then play with a friend - this is a sure-fire way to move through the ranks quickly, and without tedious grinding.