Well, since you asked about timelines, and since I myself have also taken a rather passive approach of "waiting for the hormones to do their magic..."
It took me about 2 months on hormones before I genuinely started seeing a girl in the mirror for the first time. And that was after I had gone shopping at Goodwill looking for clothes, after I had been working on my voice for a while, and after I had made a million videos of myself trying to tweak my presentation.
I finally did go out for the first time about 2.5 months into HRT. And it was just a quick little 5-minute trip where I went into Walmart in the middle of the night when nobody was there to pick up a midnight snack, and then immediately went back to the car. That gave me at least something.
For the next 5.5 months, I continued to pretty much never go out at all, only making it out in "girl mode" maybe once every 3 weeks or so, mostly to go to my therapist's office.
Finally now at 8 months, I'm part-time, and actually socializing with people and going out as a girl more often, at least several times a week now. But that is after EIGHT MONTHS of working on my voice, dieting, making videos of my walk and studying cis-women to try and make it better, still waiting for the magic HRT pills to start having more of an effect, and fighting against my own reflection every single day feeling like I'd never make it.
Even now, I'm realizing that my life has not even truly begun yet. I still only have 2 female outfits, I still feel inadequate, I still pretty much have never tried makeup whatsoever, and I am pretty much completely inept at all of the social aspects of being female. All of that isn't going to come until I get the courage to go out the door and try it. And I have to admit, I'm going to suck at it at first. I know I am. Because I have no idea what the hell I'm doing.
But it's just like learning an instrument. Nobody picks it up and is a prodigy right away. The first time, all you can make on the instrument is a few basic notes. You can maybe play something that sounds remotely like a song that you know, but it's sloppy, slow, has no technique, is unrefined. Getting better only comes through practice... through persistent practice where you're constantly working on it. And slowly, slowly, over time, you get better at it.
If you're not comfortable really getting into it, it's okay to wait, as long as you're not hurting yourself in the process. (That's admittedly what drove me to do a lot of the public pushes forward that I eventually made, was that I realized I was pretty much tearing myself up inside because I was so angry at myself for wanting to try things and yet being too much of a coward to do them. This very internal battle was the thing that has possibly cost me my job. So don't let that happen to you. If it ever reaches that point where your mind is insisting that you have to do something, do it.) But anyway, yeah, your sister is right. You really aren't going to magically wake up one day and be a girl. Even if physically you pass perfectly, there's still a million things that you have to learn in order to truly socialize as female. And trust me, those things are NOT going to come through HRT alone. Transition is a long road that you have to walk down step by step. It's not one of the beaming devices from Star Trek where you magically arrive at your destination.
So think about it like that. No, you shouldn't feel any pressure to go faster than what you're personally comfortable with. Take as long as you need to. But at the same time, if you want to get there, eventually you're going to have to start walking. Otherwise it will always be this mythical shining castle off in the distance that you can only admire from afar.
