It's hard to say. How does your mother feel about this? I think the person you should be connecting with about this, first, is her. I'd follow her lead on this.
Assure her that it's just about curiosity. You're not trying to replace her, you're not angry with her, and you aren't looking to leave her or idolize your biological father. If you have a stepfather, I'd involve him, too, and give him similar assurances. Tell him how grateful you are that he raised you. Tell them both that you love them (something that's kinda nice to do any day of the week).
One very legitimate reason for finding out about your biological father, apart from curiosity and perhaps a need for some closure, is the desire to know your biological heritage. If cancer runs in that side of the family, if you carry a genetic disease like Tay-Sachs or Sickle Cell Anemia, you should know about that, in order to make more informed reproductive decisions in the future.
Plus there is a family on that other side, but understand that you would be upsetting the apple cart, and they are definitely innocent in all of this. His other family might embrace you as a long-lost sister/brother. They might reject you. They might blame you and your mother for any problems that their parents had. They might partly accept you, and partly not. They might even ask you for money.
So recognize that this can be a bit of a minefield that you're stepping into. You have rights and you're an innocent in this as well. But be sensitive to others' concerns, needs, and feelings in all of this.
And tell your mother that you love her because, you know, you probably should.
PS Welcome to A2K.