Well, I'm a teenager, so I don't know if any advice I give you will actually help, but there's no harm in saying it anyway I suppose
I go into my 'coming out' conversations like I have nothing to lose, so I suppose in that I would be closer to someone who is out of home, like yourself. Because I don't worry about being kicked out, or of people not liking it (screw that, nobody is kicking me out of home, and if they don't like it,
they can leave, I don't need them as a friend/family member.) So I found that when I told my parents, I did so in a way that wasn't questioning. I didn't say anything like 'I think' or 'I wish to', I said 'I am' and 'I will be' etc, which let them know that while I was telling them all of this, it wasn't up for discussion. I made it clear that my mind was made up, and I was telling them so that they could come to terms with it, not so that they could question my judgement and try to talk me out of it.
That being said, I still tried to be gentle with them. I got them alone seperately, my mother kind of listened and sat there in shock, then just said, 'Okay' and awkwardly left. It took her a long time to come around, but come around she did, and we do exchange the odd sentance here and there about the topic, even though she's really uncomfortable about it. My step-father on the other hand took it quite well, he asked me a whole bunch of questions right off the bat, and didn't seem phased. It's odd, he seems to be the one who understands me the best, but he's not all that accepting. He told me my mother and he wouldn't pay for anything relating my transition, and I was cool with that.
I ended up telling my dad, who lives on the other side of the country to me, when I was down there visiting for a week, so similar to your situation in that respect

I told my aunt first, who lives with him, and then she called him into the room and I explained it to him. His reaction was a little odd. He's always loved having me as 'his little girl', so I thought he would take the news really badly, but when he found out he would be 'losing' that little straight girl in favour of a gay boy, he seemed delighted. When I asked him why, he said 'Because now I've had a daughter, a son, a straight child and a gay child. You're the whole lot in the one package.' and he seemed very happy about it. I'm his only child, so I think he saw it as a way to experience all of the possibilities of what a child could be in one go. You could mention that to your mother, it might make her smile

I made my mum laugh in the middle of the 'coming out' talk and it really lifted the tension.
Sorry I'm not an adult like you, so I can't really give any advice other than what I've dealt with, but I hope that was some help anyway

Oh, and also! Welcome!