Pokemon learn moves with types that match own because they get STAB (same type attack bonus). Also it would make very little to no sense if a Pokemon didn't have moves that were the same type as itself, otherwise what is the point of a Pokemon having a type at all.
As for Pokemon learning normal type attacks, the moves that they learn are really basic moves, and most basic moves are normal. E.g., Scraggy was born with Headbutt, which is basically another version of Tackle, & Leer. Normal isn't super effective against anything, which is just a good starting point for just hatched Pokemon, like Scraggy and Iris' Axew, so they don't do that much damage to each other when they train.
Then there's stuff like Quick Attack, which just looks cool for the animators to draw.
But I suppose the real reason is because Normal moves are the most common moves, with 167 Normal-type moves available as of Gen V. So essentially, a Pokemon will most likely have a Normal type move in its moveset. Normal is essentially the grab bag type, since a lot of moves that don't fit anywhere else are just tacked into Normal.