I actually think think that "anata no tony desu" means "I am your tony".
I am sorry if I am mistaken.
In any case, this is how I possibly could indicate my name (Suzika).
Example 1: 私はSuzikaです。Watashi wa Suzika desu. I am Suzika.
Example 2: 私の名前はSuzikaです。Watashi no namae wa Suzika desu. My name is Suzika.
I would probably use the first example more often than the other, because it seems more casual and more natural to use when introducing to friends or collegues.
When introducing myself for the first time, I would probably use a set of lines like this:
Kanji: 初めまして。私はSuzikaです。 <Next I'd probably say something like where I am from, or related to the motive of why I am introducing myself at the moment>. どうぞ宜しくお願いします。
Romaji: Hajimemashite. Watashi wa Suzika desu. < Next I'd probably ...> (Douzo) Yoroshiku Onegai (ita)shimasu.
Meaning: How do you do? I am Suzika. Please consider me positively in the future.
I haven't found any literal translation for "hajimemashte" and "douzo yoroshiku onegai (ita)shimasu", but this is the meaning that I always get when learning from textbooks, movies and translations in phrasebooks.
Here are some exmples of First Introductions:
(part 1) http://japanese.about.com/library/blintroduction.htm
(part 2) http://japanese.about.com/library/blfirstmeet2.htm
Also in "Douzo Yoroshiku Onegai (ita)shimasu", using the words Douzo, onegaishimasu, onegai (ita)shimasu, or not using them can show different politness, which is why very important to understand the Japanese perspective of politness in order to speak properly.
But I am not very secure to discuss that element.
I hope someone will clear it out for you :)
And I hope I might have helped.
(Please keep the hard work).