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There doesn't seem to be much in the book that is really outdated; mechanics hasn't changed much since 1967. However, I would not recommend this book for self-study. It doesn't have any homework problems, and has very few examples. For self-study, you really need a book with problems as well as answers and/or complete solutions (or at least answer checking on a computer).
I don't know what your background is, but the book seems to assume strong previous knowledge of vectors, including dot products, which most students in the US don't have. In general it looks like a book that's meant for people who have already taken a really solid high school course (i.e., something much more substantial than what is offered in the US except in an AP course).
As a matter of taste, I strongly dislike the way the principles are presented without motivation or connection to experiment, e.g., the definition of work without any physical motivation or interpretation.
What's the most recommended first course in physics?
It depends completely on your level. If you're at a level where you can make sense of the Landau book, then I would suggest Kleppner and Kolenkow. If you've got no money and no access to a university library, there are other free physics books online. I maintain a catalog of free books here: http://theassayer.org