There are probably as many ways to do it as there are people transitioning.
Social transitioning is basically just starting to live full-time in your new gender role. It implies that you are out to everyone (or alternatively, that you cut all ties to your old life and start to live 'stealth' in your new role). So part of the process is coming out to everyone you know and everyone you interact with. It also involves presenting in your new role, whatever that means for you in terms of clothing and appearance.
For me, it meant coming out in person to several community groups that I am a member of, and coming out by email to others. To avoid offending people, it helps to give some attention to what order you tell people. I came out to the (mostly) women's group first, since I would need them as allies. But before I did that, I talked to the Chief of the Fire Department (I/m a member), since I knew he would find out by gossip if I didn't. You don't want people you need as allies to find out by gossip.
By the time I did all that, I had already been on HRT for several months, and my figure was coming along nicely. I was able to do an overnight transformation simply by switching from a compression bra to a padded bra. I also had a decent wardrobe assembled, and I had been getting electrolysis treatments on my face. I had taken some speech lessons and was starting to get the pitch of my voice higher.
After telling the women's group, I immediately began presenting as female. So when I told the Fire Department several days later, I showed up in girl jeans, a girl T-shirt, and a wig. I got some strange looks and comments, but I had planned it for the monthly meeting, and I had requested time on the agenda to make my announcement, so all became clear fairly quickly.
Working out the logistics of whom to tell when actually occupied a significant amount of time in the weeks prior!
From that day onward, I have presented only as female. I had started using women's washrooms earlier, when I was presenting part-time, but, as of starting full-time, I have used nothing else. Going grocery shopping, taking the car in to get serviced, whatever, I present female. If I have a three-day stubble because I have an upcoming electrolysis appointment, I still present female. Yeah, I'm a girl with stubble, it's part of the deal.
So, in summary, social transition is the process of altering your presentation to the desired role. I kind of glossed over part-time presentation, but that is a big part of social transition: learning your presentation in preparation for full-time.
And there is also the legal / bureaucratic stuff of getting your name and gender marker changed legally and on all your accounts.
