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.
I never bothered setting up an LLC or business entity. My income would all pass through for tax purposes anyway, so I operate as a sole proprietor. That requires no setup. I never had a customer balk at paying me directly. It does mean they have to file a 1099 with the IRS, but if they won’t you can still report the income. If you have employees or more complex business needs sole proprietor might not work for you, but it remains the most common form of running a business. Since 2013 I have worked through an agency, they handle billing/payment for those customers and pay me by direct deposit.
Unless you need a merchant account or have complex accounting requirements you probably don’t need a separate business bank account. You can set up separate accounts for personal and business use, to separate the transactions. I just used one account and kept track of business deductions with a spreadsheet and folder of receipts — freelancers usually don’t have a lot of deductible business expenses or things like leases, inventory, depreciating assets, etc.
I use Charles Schwab Investor Checking because they refund all ATM fees, which can add up when traveling. They also have excellent customer service, for example sending a replacement for a lost debit card to Bangkok via FedEx. I advise keeping two accounts (in case you lose a debit card) and a couple of credit cards you auto-pay. Pro tip: if your US bank ever asks if you live outside the USA, tell them no. They don’t check but the law requires them to ask, and if you say yes they will either close your account or change it to a non-resident account.
I used Traveling Mailbox to get a Washington address (no state income tax) and never had any problems. They scan your mail and you read it online. They will forward if you need that. They will deposit checks too, for a fee, though I used my mobile banking app to deposit scans of paper checks from my laptop screen.
You probably need a physical residential address in the US to open bank accounts (I used my daughter’s address) but once opened you can change the address to a mail receiving service.
A US drivers license comes in handy overseas should you need to drive or rent a car. It also establishes state residency for tax purposes. I chose Washington, Nevada works too, easy and fast to get a license. You need a residential address for the DMV to mail the license to, so family, friend, Airbnb, even a hotel.
The IRS doesn’t care where you live or what address you use as long as you file and pay. I used my Traveling Mailbox address with them.
Most people can reduce and nearly eliminate paper mail by choosing paperless banking and credit card payments. After a while I got very little important postal mail. Most of my customers paid by wire transfer or direct deposit/EFT to my US bank, or via PayPal or Transferwise. A few still used paper checks in the mail as mentioned above.
Because I manage infrastructure (cloud hosting) for multiple customers I needed a US phone number that didn’t change as I moved around. I used Google Fi for that, in a phone that supports two SIM cards. Newer iPhones have a virtual SIM and a physical SIM slot. If you stay in one place for a while a local SIM is cheaper and more useful so plan for that. I met a few nomads who had two phones, one with a US SIM, one for local access. My phone also worked great as a wifi hotspot — cellular internet service is cheap in a lot of countries, cheaper than using Fi anyway.