When is it OK to introduce non-children's music?
Seriously, there is absolutely no reason to "ease" your kids into music gradually.
We started our son our with everything from John Williams, to Frank Sinatra, to The Ramones, starting before he was born and continuing right up to today (he's three).
We mixed in children's music (for example, there's a duet album by David Grisman and Jerry Garcia playing a variety of children's songs that all three of us loved to sing along to), but he only really started liking more "traditional" children's music, and asking for it specifically, when he hears it on the radio, youtube, or during a TV show or movie (which is why I am utterly sick of any song from Frozen at this point!).
My son, at this point, happily listens to, or even sings, a variety of music from songs from Cars and Frozen, to silly songs like The Duck Song or The Toast Song.
There's even been some (controversial) research onto the impact of "adult" music on children. See The Mozart Effect for a classic example (sorry for the pun!).
However, if you start your child off on music geared towards children, and only later introduce "adult" music, you may meet with some resistance. Children (at least mine) tend to like what is familiar, unless convinced otherwise.
Some children's radio stations (we have XM radio, and listen to a channel that I believe is called "Kids' Place Live") do a decent job of mixing in a fairly wide variety of songs, from classic standards that are kid-friendly, to current (and not-so-current) Disney soundtracks, to age-appropriate pop-songs (it was our introduction to "What Does the Fox Say?", which my son finds hysterical).