When I play sports, I get shortness of breath really quickly. Is this normal?

No, check w/ your MD. It is prudent and recommended that you obtain adequate medical clearance before partaking in sports. Many respiratory ailments (sinus conditions, asthma, and allergies, to name a few) are easily treated and manged if adequate medical advice is obtained.
Needs throrough exam. There are a number of conditions that can cause shortness of breath. Detail history and a thorough physical exam by your doctor will sort out the cause(s).

Related Questions

I get chest pains, shortness of breath, wheezing and constant coughing after playing sport. I exercise 3 times a week, and play soccer on the weekend?

Asthma. It sounds like you have asthma, you need to see your doctor for an inhaler to use before exercising, and you need to check your lung function test to determine if you need a maintenance inhaler. Read more...
See your doctor. You could have exercise induced asthma or an allergy induced asthma. You should see a doctor for a diagnosis and treatment. Read more...