I get this fluttering in my chest and sometimes my heart races for no reason. I am only 24. What could it be?

Rhythm problem. It is common even in young healthy persons for the heart to suddenly skip, beat to fast, or beat irregularly. Although this may be a disturbing symptom, most of the time in people your age, it is benign. That said, you should see a doctor to get an EKG and confirm exactly what kind of irregular heart beat you are having and to see if treatment or further testing is needed.
PSVT. There are many possibilities but, statistically, the most likely diagnosis in a healthy young woman would be paroxysmal supraventricular tachycardia. A 24 hour or 30 day event monitoring exam would provide an accurate diagnosis. There are many treatments depending on the cause.