We, as viewers have never heard Gallifreyan being spoke. The best we've heard is the names of a few people (Romanadvoratrelundar), places (Gallifrey), etc., and even those may be "half-translated". Some of the Big Finish audios have featured brief snippets in Gallifreyan, but not nearly enough to figure out the language. Some of the novels—especially the Virgin New Adventures—have had more extended snippets of Gallifreyan, transliterated into Roman letters, but it's still not nearly enough to figure out the language.
The reality is that, unlike Klingon or Quenya, no one ever sat down and designed a Gallifreyan language to use on the show. The farthest anyone has gone was the editors of the New Series Adventures, who came up with a set of numerals (which are used in a base-10 positional system, just like our Arabic numerals), and the pronunciation for the Gallifreyan numbers 1 through 9. Other than that, each time a writer needs a bit of Gallifreyan, he just invents some nonsense that sounds alien. Every once in a while someone remembers a previous phrase or name and reuses it. But that's it. So, there is no language to learn.
If you look on this wiki at the entry for "Gallifreyan (language)", the ONLY on-screen canon spoken Gallifreyan word we have ever had is "valeyard" - "learned court prosecutor".