This is a big open-ended question for Americans who are permanent digital nomads (or who choose not to have a permanent address in the US).
There are many choices to make, and the roadmap is unclear. Searching the web predictably turns up a lot of SEO-game-quality results, so it's difficult to know what is good advice.
For those who have experienced this, please share what you are doing or have learned.
Mailing Address
There are virtual postal mail services which can be used as a "permanent" mailing address. (A USPS mailbox cannot, because they require two limited forms of ID; for those of us who do not/no longer have a US drivers license, our only accepted ID is a passport.)
Business Formation
LLC seems to be the wisest choice for those who do contract and consulting work, particularly for US clients. There are many states which seem to offer attractive benefits as an LLC location, such as Wyoming, Delaware, Nevada, and Texas. I list them in that order because from my reading that is the best order, but perhaps you have more experience and can share?
Taxes
One of the joys if being an American is that we have to file taxes for the US regardless of where we are or where we are earning. There is no opt-out short of giving up citizenship (which obviously requires having another citizenship/passport in another country). In my case, having Dutch residence, I have to file and pay in Netherlands and also file in US. (There's no double taxation, so paying more taxes in NL means I have zero taxes due for US... but I still have to file; that will never go away.)
How do you manage this? The temptation is to only have one residence - US (virtual mail address), then stay in other countries only as allowed by visa limits. For Schengen, that's 180 days per year -- enough if you're a full time nomad.
Banking
What bank(s) do you use? Wise (formerly Transferwise) may be a viable option for receiving money from clients in the US and elsewhere. HSBC has been suggested, but the details of their account setup requirements seem not to apply.
Experiential advice on any of these topics is highly appreciated!
For example, residency requirements vary from state to state and a virtual post office box might not meet residency requirements for any given state.
Good luck.