"I will get a haircut today" means that someone will cut your hair for you. "I will cut my hair" implies that you will do it yourself.

'I will get a haircut today' is correct (you will go to a hairdressers and they will cut your hair.) OR 'I will get my hair cut today' (notice the difference, 'haircut' is a noun, whereas in this sentence you are using 'cut' as a verb.) 'I will cut my hair' is a correct sentence, but it means that YOU will cut your own hair. I think it's unlikely that you mean that.