I am not a laywer - since you are about to start a business (even if just 'for fun'), you really need talk to one.
What about small indie developers? Do they simply trade mark their studio name and have no registration with the government?
You have to register trade marks, and they cost about $275+ apiece in the USA.
Also, what some indie developers do out of ignorance, and 'seem to be okay', doesn't mean it's a good thing to do.
Or do they at the least register a business under sole proprietorship / partnership?
You actually don't have to register sole proprietorships, as far as I know (which admittedly isn't that far), but you can optionally register them.
However, they don't offer you any protection. If your game software accidentally wipes out my harddrive, and I sue you, even if your EULA says, "No responsibility for damages caused to your PC", EULAs haven't actually been proved to be legally binding in court, so I can sue you anyway, and if you lose, I can take your house and any other possessions you own, even the ones unrelated to your business. This makes you a good target for lawyers, knowing you have money, and knowing you are vulnerable to lawsuits.
However, if you're a LLC (Limited Liability Company) or a LLP (Limited Liability Partnership), your legal losses (liabilities) are limited to only what the business owns, and not what you personally own.
Again: I am not a laywer.
I'd figure out if I am doing it for a hobby, or for extra income on the side, or as a business. That'd change how I'd approach it.
As a hobbyist, I wouldn't bother with any of it.
As extra income on the side, I'd just take some small precautions (maybe an LLC but not bothering with trademarks).
As a business, I'm personally planning on doing first an LLC, then later expanding to a Series LLC with several sub-LLCs, for segregating separate parts of my business once things get underway. But I plan to talk to a lawyer once I get closer to release.