I have two little kids (I know that's not three) and public transit or biking (I can fit both on my bike) is great for outings. It turns the journey into an experience itself. Driving with the kids strapped into car seats the whole time is much less fun. I live in a very bikable town within walking distance to light rail, but that was by design. We bought our house by drawing 1/2 mile circles around parks, rail stops, and grocery stores. Bought the first place that intersected all three. I was just thinking recently about how my kids likely be autonomous much earlier. By age 11 or 12 they'll be able to go all around the city on their own.
Curiously, where do you live? I wouldn't let my kids ride transit by themselves; that's where the crum bums go to pee.
I also don't want to be against a grocery store as the noise from the large parking lot (and busy road) is annoying. Living by a park is highly desirable.
We're protected from the busy road and the grocery store parking lot by elevation; we're 50 ft below either. We can hear the highway, but I only notice it when the din is missing when we visit friends in the country!