So i have a 32 gauge H&R dart gun. is it possible to fire just normal shots out of it?

What do you mean by normal? For your 32 gauge, the 32 gauge shot would be a normal... For my 12 gauge, the 12 gauge shot will be normal. There is that thing caliber, you know. where the hole of certain diameter does fit only to rounds of the same diameter? Have you study geometry by any chance? I think they have something like that in high school.