I have never transitioned from one to the other, but I did my undergrad at at University on the quarter system. I'm not sure how much it will matter for your decision making process because you will only be taking classes for a year or two.
But some things to consider about the quarter system:
Each quarter is about 10 or 11 weeks (depends on where you go). This means that midterms come up very quickly, usually the first round is about 2-3 weeks in. They sneak up on you so if you've been slacking off, you will find yourself cramming right before the exam. However since there is less time in between each midterm, there is less information to learn.
Also, if you don't like a class, its over in 10 weeks and it won't drag on forever. Also christmas break and spring break are the transitions between the quarters so you don't have to worry about studying over the break to take a test when you get back. You get to enjoy your breaks and then start a whole new round of classes after the break.
There are pro's and cons to each I'm sure. I liked the quarter system, and i'm sure it will take me some getting used to the semester system wherever I end up for grad school. Good luck!