The next time it happens, try
to move around a little (or if
you are about
to faint, sit down). When
you move around,
you prevent blood from being pooled up in your dilated veins (which happens after exercise), you squeeze the blood back to your heart (where it can have useful purposes).
I assume you aren't dizzy/chest pain/fainting DURING exercise. If you are, that's far more serious and should be evaluated ASAP.
The changes in heart rate with breathing is completely normal.