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When in the main screen after log in, in the upper-right, there is a button with a icon that looks like the head and shoulders of a person. Click that to go into your Summoner Profile. In there, you can see what champions you own, set up Rune Pages and Mastery Pages, view your recent match history, etc.
On the Champions tab, any champion whose icon is colored in is one available to you currently. If they have a little exclamation point in the corner, that means they are free-to-play that week (Tuesday rotations on those), and thus if you have not purchased them, they will become unavailable the following Tuesday.
Summoner Spells unlock as you gain levels on your account. You'll have the last one unlocked once you hit level 12.
How do I check what champions, spells etc are available for me?
New to this