Remove Author, Post and Category-related Text from a Theme?

In functions.php find:

 if ( ! function_exists( 'twentyten_posted_on' ) ) :
/**
 * Prints HTML with meta information for the current post—date/time and author.
 *
 * @since Twenty Ten 1.0
 */
function twentyten_posted_on() {
    printf( __( '<span class="https://wordpress.stackexchange.com/questions/6277/%1$s">Posted on</span> %2$s <span class="meta-sep">by</span> %3$s', 'twentyten' ),
        'meta-prep meta-prep-author',
        sprintf( '<a href="https://wordpress.stackexchange.com/questions/6277/%1$s" title="%2$s" rel="bookmark"><span class="entry-date">%3$s</span></a>',
            get_permalink(),
            esc_attr( get_the_time() ),
            get_the_date()
            ),
        sprintf( '<span class="author vcard"><a class="url fn n" href="https://wordpress.stackexchange.com/questions/6277/%1$s" title="%2$s">%3$s</a></span>',
            get_author_posts_url( get_the_author_meta( 'ID' ) ),
            sprintf( esc_attr__( 'View all posts by %s', 'twentyten' ), get_the_author() ),
            get_the_author()
        )
    );
}
endif;

Change to:

if ( ! function_exists( 'twentyten_posted_on' ) ) :
/**
 * Prints HTML with meta information for the current post—date/time and author.
 *
 * @since Twenty Ten 1.0
 */
function twentyten_posted_on() {
    printf( __( '<span class="https://wordpress.stackexchange.com/questions/6277/%1$s">Posted on</span> %2$s <span class="meta-sep">by</span> %3$s', 'twentyten' ),
        'meta-prep meta-prep-author',
        sprintf( '<a href="https://wordpress.stackexchange.com/questions/6277/%1$s" title="%2$s" rel="bookmark"><span class="entry-date">%3$s</span></a>',
            get_permalink(),
            esc_attr( get_the_time() ),
            get_the_date()
        ),

    );
}
endif;

In other words you’re just removing the bit of code that reads:

sprintf( '<span class="author vcard"><a class="url fn n" href="https://wordpress.stackexchange.com/questions/6277/%1$s" title="%2$s">%3$s</a></span>',
                get_author_posts_url( get_the_author_meta( 'ID' ) ),
                sprintf( esc_attr__( 'View all posts by %s', 'twentyten' ), get_the_author() ),
                get_the_author()
            )