How do I selectively sync music to my iPhone?
If I were you, I would pay the $25 a year for iTunes Match. Provided you all share a single Apple ID, this will let all of you use it's features. 
The one you are interested in is


Turn on iTunes Match (having synced your computer iTunes to Match already)
Make sure "show all music" is on
Go through and download the songs you like onto your device. 
Turn off "show all music" 


With that you have only the songs you care about, don't need to even see your kids music, and can manage it all from your phone. 
i am an iTunes Match power user, and there are dozens of other ways to use it, but that is a whole different question.