Hello and welcome to iMore! It's nice to have you here. I've never had a BBM so I can't speak to your experiences with a BBM, but I can hopefully point you in the right direction for what you're looking for on your iPhone.
In order to set up notifications, you need to go to Settings>Notifications>Calendar. From there you can set up your desired notifications, including banners, sounds, and badges.
This is a link to an article titled How to customize your Calendar settings on iPhone and iPad. Hopefully, it will help you out.
And finally, in order for us to continue to help you out, you may want to consider setting up an account here on iMore.
Here's how.