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

The use of the TM symbol is the initial step in securing TM protection - it effectively tells the world that you are currently using the mark as a TM and that you are pursuing TM registration which you should pursue by filing a TM application. The application will be approved for registration or rejected depending upon several factors, such as, for example, another party may already have a registration for the same or similar mark for the same or similar goods, or your mark may be considered descriptive, or suggestive. You should consult a TM professional in order to pursue your TM rights.

You need to file a trademark application with the US Patent and Trademark Office. It's a good idea before filing the application to have a trademark search done to ensure that there is not a prior user with a confusingly similar mark.

In the US, you obtain a trademark based upon use or the above federal registration. Trademark owners generally use the TM prior to the registration, when they can use the (R).

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.