first generation would probably never learn to have a heart because the only people who survived to be able to have children are most likely heartless killers.... the children would learn how to survive in this environment and pass down this knowledge generation after generation and refine living during the zombie apocalypse to its fullest capacity.
The simple truth is they wouldn't grow up spoiled. They would value their family and friends more and be colder to those that aren't. They would move over and beyond the bad things to focus on the truly important things like food, community, and survival. And I doubt they would buddy fuk people over at the drop of a hat like most everyone does in this day and age, just for the "mighty" dollar or whatever they happen to want at the time. Sounds a lot like how my parents, grand parents, and their parents were. You know, those people who lived through and during such fun times as WW1, The Great Depression, and WW2 just to name a few.
Oh, and sure there would be a few bad apples just like with most everything else, but bad apples make great fertilizer =) As for heartless killers well they maybe a bit more liberal with killing but in the end I think it wound be more out of necessity then heartlessness.
i think they would grow up fleeing for survival but as was stated by others they would care for those around them probably much more so than you would imagine but be able to flip that on and off when the time comes to kill an infected member of your family or community
They may be more caring then we are, but it would not be idiot compassion.
And the Spartans can be looked up to, but they may not be the best example. They were a slavery society in which considerable time and resources were devoted to keeping the helot (slave) majority sufficiently terrorized. This was one of the reasons causing spartan society's downfall.