Normal Elo is 'hidden,' which means you will never get to see it unless you work for Riot. The reasoning behind this is to make it a more 'casual' game where you're not stressing over numbers.
If you want to be competitive, you play ranked games once you are ready (you can play them once you reach level 30, but just being 30 doesn't mean you are ready). Ranked games have their own Elo, separate from normal Elo, but this Elo is visible to you (and others if it's above a threshold). You might wish to be competitive before then, but you really won't be experienced enough at the game until a while after you hit 30 to be truly competitive.