I'm not a young mother, but I can help you out.

Measles can still be gotten. The vaccine that had been used since the '50s turns out to wear off. The CDC and others have been recommending adults who got the vaccine from like the mid-1950s through the 1970s (at least) get revaccinated. I had to do it when I went back to college at 30, believe it or not, or they wouldn't let me take classes.

There is now a "MMR" vaccine, measles-mumps-rubella, but I'm not up to date anymore on the ages kids are supposed to get it.