As you begin, you'll notice that there are two sets of numbers at the top of your profile screen. One is RP (Riot Points) and the other is IP (Influence Points). RP is accumulated at the Riot store by spending cash. It is not recommended dumping cash into this game at an early stage. There is no real reason to dump cash into this game at all, unless you have cash to burn. Everything you need can be purchased with IP. IP is accumulated by playing games. Winning games accumulates more IP than losing games. Playing against human opponents accumulates more IP than playing against bots.
You may be tempted to spend these points testing out runes, or buying champions. It is recommended that you save your points. There are free-to-play champions that you can select from. The free champions rotate on a weekly basis, so you can always play something new. By playing all the champions, you get a feel for what other champions are capable of, as well as find what you're good at.
There are Tier 1, 2 and 3 runes. These runes enhance your game by giving you added benefit. You may be tempted to buy Tier 1 and Tier 2 runes, but these are garbage. Once you hit level 20, you can buy Tier 3 runes, which will stay with you for life. The Tier 1 and Tier 2 runes will cost you way too many IP and will be worthless once you hit 20. Save your points. You'll be thankful once you hit 20. Once you have a full set of runes for an AD (Attack Damage) champ and an AP (Ability Power) champ, you can start purchasing your favorite champions. Some people also have a rune page for a jungler as well as for their tanks.