Can I have sex without condom if on pill?

So I just started taking the pill on Monday because it was the first day of my period, and I've been taking it on time every day since then. My period is finally over and I was wondering if it would be safe to have sex without a condom today ?
I have read that if you start taking it on the first day of your period that you are automatically protected and don't have to wait a week like you would if you were to take it on any other day
Not sure if that's true so thought it would be best to ask and get opinions