Whenever I blink, I feel pain in my eyes. Is this normal? Will it go away by itself?

Depends. Pain with blinking usually means there is a disturbance on your cornea. This could be from relatively simple conditions such as dry eye or blepharitis (eyelash inflammation), or more serious conditions. If the pain does not rapidly improve with simple artificial tear placement or contact lens discontinuation then seek ophthalmologic care.
See an eye doctor. Pain when blinking can be from something as simple as dry eyes, to a stye or blocked oil gland in the lid. There can be other causes as well. Constant discomfort with blinking should not be considered normal. See an eye doctor; your condition can usually be treated very easily once the cause is determined.