If I want to become a physical therapist that specializes in sports, where should I begin?

My best recommendation would be to study athletic training as an undergraduate. That degree would clear the majority of your pre-requisites for graduate school (with the exception of Physics and possibly chemistry.) You'll get a LOT more detailed instruction on anatomy, kinesiology, rehabilitation, special tests and the field of healthcare in general. You'll also complete several externships where you work under licensed athletic trainers in a variety of settings -- this should increase your exposure and professional network. Most of all, you'll be able to be a licensed athletic trainer ("ATC")

Then, continuing on for your doctorate in physical therapy will open more doors and increase your earning potential beyond what either degree would on its own. Earning a bachelor's in athletic training shouldn't take any longer than a "traditional" bachelor's degree.

That is hands-down the best option IMO. (And frankly, you could land a dream job before graduate school and not even need to attend or save for a few years and be able to graduate without the average $96,149 in debt. http://www.apta.org/Media/Releases/Legislative/2011/4/8/) After graduation from physical therapy school, you can also pursue specialized credentials as a Physical Therapist within sports medicine (see https://spts.org/) but if you're an ATC, there really isn't any need.

RESOURCES:
Finding ATC Schools, click the line below "Don't have a Bachelors" https://www.nata.org/about/athletic-training/obtain-certification
Finding PT Schools, as well as their prerequisites, can be done here: PTCAS.org