I'm trying to create a character. I've already worked out some of my character. My character is, at the moment, superstitious with that being pretty much the only personality trait. It has glamours and can change shape. I'm also not planning on following a template. I was hoping to have my character get magic through exposure, I guess, but I'm not quite sure how a character is exposed to magic in a way that would cause the character to be magical. How is a character exposed to magic?
A lot is going to depend on what you mean by 'get magic'. A lot of it is going to depend on your GM, so that would be a good first place to start, i.e. talking with him about what you're trying to do. Moving on to the character, there are two sides to this, but after the comment above, I'll start with systematically.
In DFRPG, you have to justify supernatural powers with, at the least, an appropriate High Concept (see
YW146 for a comparison between Supernatural Powers and Mortal Stunts). So whatever the character was exposed to (whether it be some Thaumaturgic ritual or an artifact of power) would dictate what this High Concept was defined to be. That change in High Concept is the first step to work out with your GM (and requires a Major Milestone as noted by @justin-time below).
After that step, you'd have to work out with your GM what powers are available based on what the triggering magical event is. If the trappings of the magical event are going to make the character a spell caster for all intents and purposes, a look at sponsored magic would be in order.
The larger issue (and the reason I stated that you should talk to the GM) is the in-game rationale, especially if you're planning to do this during play. It is going to depend a lot on your GM's interpretation, as such an event isn't really covered in the books as written. Find out before you get too deep into the concept what your GM will allow, as that's going to be the framework for if this is possible, and under what circumstances it would be possible.
In a game as abstract as FATE, you're actually facilitated fairly well as to how these sorts of things may happen. If you can feed a pretense into the plot device, and your GM is okay with the change, the rest become plot bait and you crank the device. Popular examples involve rare (nay unheard of) rituals good for just one more use, untapped latent power, finding a place of power, finding a being willing to "Sponsor" you and have the actual magic develop on its own over IC time, and my personal favorite: the body swap where the minds change but not the powers.
Your patience is entirely critical, because most of the time you won't just simply become a full blown Wizard overnight. You may have to do some fledgeling work, then Sorcerer, Conjurer, and finally get chest deep. A sudden change requires a mega burst of DM Magic and heads turning the other way even in the buttery soft sci-fi world that is Dresden.
Aside from Game Rules as Written (which would pretty much state that if you have the Refresh to spend you can gain the power) from a Dresdenverse Standpoint, the PC will need to be able to first sense magic and it's effects (usually granted by The Sight) and then have the metaphysical power to power a spell (have the Conviction skill at a level of at least +1 or higher).
Think of The Sight as the sense that lets you perceive magic. Asking a Blind person to pick up a gun and shoot at a sight orientated target is a pretty tall order. —Sure, a blind person might be able to fire a gun at a target and hit it, but it's highly unlikely and has a high likelihood of failure. The same is true for Magic, without a sense to guide its use the success rate is terribly low. The extra caveat with Magic being the more failures you have with attempting to use it, results in self doubt about your ability to continue to call up power. Each failure resulting in less and less power to channel until you don't have the self belief (conviction) to cast the spell.