Yeah, magic would be learnable, in theory by everyone, but much like other studies your natural abilities would have an affect on how much and how complicated the magic you could learn would be.
Average people would be VERY limited in what they could affect, and how much power they could control.
The masses though COULD interact with other magic that was set up by specialists or "wizards" who have much stronger potential, which could allow you to for instance control magical carts to move around with etc.
Nobody would have the power to blow up the world, and even the strongest would only be able to possibly level a city at most (and that would wipe them out and make them completely vulnerable.
Things like flying though would be possible, teleportation as well, you could create things out of thin air (though it would require energy of some form, so transforming something would be far easier/safer).
There would be rules to magic, but society would not have developed scientific method so those rules would be mostly unknown (other then basic rules).
So basically you're asking if I want to live in Harry Potter world. Pass. ---