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.
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.