Registering a trademark in the US/using "TM symbol"

The two public answers you have received thus far are pretty complete. I would just add that you could review the U.S. Patent and Trademark Office website at www.uspto.gov for some videos and additional information on U.S. trademark law, as well as an explanation of the U.S. trademark registration process and trademark application forms if you dare to try filing the application on your own. One of the videos should explain the TM and R in a circle marking systems as well. When Mr. Alderman said, "Trademark owners generally use the TM prior to the registration, when they can use the (R)," I think he meant to say that "trademark owners use the TM..., when they canNOT use the (R)" because the (R) or R in a circle is only available after a mark is federally registered. The TM symbol is always available and recommended to alert the public that the mark is in use in U.S. Commerce and common law trademark rights may be claimed. In the U.S., a business can acquire trademark rights merely by using the mark in commerce and one does not need to register the mark to claim rights in it. Although trademark registration is strongly recommended to be able to claim nationwide rights in the mark.
Moreover, since one can accrue trademark rights merely by using the mark in U.S. Commerce, it is important that a business search the availability of the mark in the U.S. prior to using any mark in U.S. Commerce, both by what is registered or applied-for in the USPTO and through an internet search or the like, because any prior used or federally registered mark could allege trademark infringement for the subsequent use or registration of your mark in the U.S.
You are best advised to consult a trademark attorney which could be a trademark attorney in Europe who will likely maintain a network of trademark attorneys around the world, including the U.S.