Trademarking a name

You can put a tm symbol immediately on anything you want to claim as a trademark as well a publish it in print as evidence. But formal filing, USPTO.gov, will certainly aid in court.

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Proof... page one of pdf from basic facts at USPTO.gov

SHOULD I REGISTER MY MARK? Is registration of my mark required?

No. You can establish rights in a mark based on use of the mark in commerce, without a registration. However, owning a federal trademark registration on the Principal Register provides several advantages, including:

• Public notice of your claim of ownership of the mark; • A legal presumption of your ownership of the mark and your exclusive right to use the mark nationwide on or in connection with the goods/services listed in the registration; • The ability to bring an action concerning the mark in federal court; • The use of the U.S. registration as a basis to obtain registration in foreign countries; • The ability to record the U.S. registration with the U.S. Customs and Border Protection (CBP) Service to prevent importation of infringing foreign goods; • The right to use the federal registration symbol ®; and • Listing in the United States Patent and Trademark Office’s online databases.

If it's really unique can I trademark the name in general or still have to specify what it sells even if it's unique?

Probably depends on which country you're trademarking in, and you didn't say. For the US and EU you must specify which areas of business you're in. (This applies to registered trademarks, which seems to be what you mean.)

And can I trademark something myself?

Yes. But a lawyer who specializes in trademarks need not be that expensive, and knows the finer details.

But really, see the past discussion about this topic, i.e. the "Trademark" tag, and here and here and here.