I'm glad to hear the talk with your friends went pretty well! As for how to come out to your mom, there's no need to apologize for asking those questions. Making that choice can feel pretty intimidating, even when you feel it's ultimately the right call, and it's great that your asking questions to help you feel more prepared for it.
As for how to approach your mom, that will depend a little on her personality and what you know about how she tends to react to things and what ways of presenting information work best with her. We actually have an article that covers coming out as trans that you might find helpful right now: Trans Summer School: Let's Bust Out of This Closet!
. There are a few things you can do to make yourself feel more prepared for this conversation (which can help you feel calmer about the whole thing). One is to sit for a bit and think about what questions you think your mom is likely to have about your statement, and whether or not you want to be the one to answer all those questions or if you'd prefer to give her resources to help her learn on her own time. We even have a piece for parents on how to be supportive when someone comes out as trans, and I'd be happy to link you to that if you'd like to give it to her. Another thing to consider is if there are any particular things you want her to do as a result of you telling her you're trans. Those things could include calling you by a different name and pronouns, or helping you out with a specific situation (for instance, some people want or need their parents to help them access clothes that feel more in line with their gender). Thinking about those steps and the steps in the article, are there any that feel really difficult or intimidating to you? Or do they mostly feel doable?