I keep feeling weird fluttery feeling. How can I tell if this is a serious problem or not?

Things to remember. You can be somewhat reassured that fluttering in the chest, even if due abnormal heart rhythm, is very rarely a serious problem. While rhythm problems can cause a lot of symptoms, they rarely are potentially dangerous or life-threatening. One should try to correlate symptoms with an actual pulse irregularity, either by simply feeling your own pulse at the wrist, or by wearing a portable monitor.
See a doctor. Fluttering sensations commonly mean the heart rhythm is irregular. If your pulse is not regular when you get the symptoms, that would prove it. An EKG during the symptoms would be even better. Sometimes the symptoms are hard to catch so we put a monitor on the patient which he or she wears for a whole day to see if the heart is indeed skipping. Fluttering can occur even without skipping.