The way it always worked for us was that the hiring manager would announce the arrival of the new person via email. Everyone would email the new person and say hi - those who would work closely with them would email their own intro. And their teammates would probably call them just to say hi. All the new person has to do was sit back, let the intros come in, and respond in kind.
I should add that this wasn't a formal process. It just developed into a part of our culture over the years.
Hopefully, the team you are joining is similarly friendly. (Frankly, it is a red flag if they are not) Remote teams don't get the little face-to-face interactions that co-located teams do, so their phone calls tend to have more non-work talk to make up for it, and you get to know people just by being on the team and communicating.
It takes longer to execute. But it's worth taking the time now, upfront to start building a relationship. Show an interest in the person, what their working on. ASK if there's anything you might help them with?.
I had a colleague from Sao Paulo start that way. We became fast pals.
Introduce your job role, where you've come on from, what you'll be working on, who you report to, etc, maybe mention some hobbies or some extra unique fact about you. I'd keep that last part to 1 or 2 points but I think the key is to come up with a future talking point when you're speaking to people for the first time.
From the cultural angle: I don't know how many Americans you know but I generally find them to be quite a friendly bunch so I'm sure whatever you say will be appropriate so long as it's positive and upbeat.
Schedule chats with everybody you can.
Make sure to use a profile picture of yourself on all the different apps you use to communicate.