I don't think there is any right or perfect way to do this. I had various approaches.

At work, I just put a message in a large group chat with all my colleagues. I think I said something like "I've been in therapy and have come to realize I'm trans. My new name is XX and my pronouns are YY". I always told people I was happy to answer questions, but almost nobody wanted to discuss it (I found this especially true the more they were identified as an "ally").

I think with my family, I just called them up one by one and just said, "hey, I've realized I'm trans and I'm transitioning, my new name is XX and my pronouns are YY. Just wanted to let you know." etc.

With friends I usually just texted the update. I socially transitioned all at once, and in every sphere of my life - so I didn't have some people knowing and others not knowing. I found this easiest and reduced the stress of people "finding out" and so on.

It doesn't have to be awkward - if it helps, just think: what do they really need to know? People want to know how to be respectful or polite, so it's important they know what name to call you, and what pronouns to use for you. They might also want assurances or to know that if they make honest mistakes they won't be punished or categorized as "transphobic", etc. Usually I found it easiest to assuage these fears by insisting I didn't expect perfection and that it was just important to me when there is a mistake that I know it's a mistake and not an intentional misgendering or deadnaming, and that it's similar with gender neutral language like "guys" or "dude" - as long as I know it's intended in a gender neutral way and not intended as misgendering, it's all good.

One person told me they were really nervous at first because they felt they didn't know how to treat me as a trans person, but that it was all made easy when they realized I was just a woman and that all they had to do was think of me as a woman - they said that was so easy to do and it really made them feel comfortable because it wasn't such a mystery "what to do" anymore.

The main struggle they had after then was just knowing how to bring me up with mixed company who may have known me before I transitioned and not heard I transitioned, and what they did was just refer to me by my new name and pronouns, and if there was visible confusion just saying "oh, person X was X but now they're Y". Since I was open, I didn't mind that this was handled.