One of the previous posters was correct that it all depends on who you ask. I have been a student of the Bible for the last 15 years and I've come across many different views.
Many mainliners would believe that children are born innocent and remain that way until they reach the "age of accountability." Although the Bible never addresses such a notion they believe that children or the mentally retarded are not held accountable for what they do not understand. They believe that for the rest of the population that on the one hand we are all born sinners who will perish in Hell if we are not saved, but on the other hand they feel that children and those who cannot understand are exempt from that status. Or in other words, even though they are still technically sinners that God will show special grace to them because "he understands."
There is also a group of "Reformed" folks who believe that even though God only has a select group of people who will be "saved" that children are also somehow exempt. They would believe, in so many words, that God has already chosen who is going to Heaven before they make the decision to follow him or not. Some would even go a step further to believe that God has already selected both who is going to Heaven and who is going to Hell...but even if you are already destined for Hell that you are still responsible for your actions.
When I was a Christian I was not totally sure which view to take. I just settled with believing that if you were "saved" then you knew it and if you weren't then you would find out in Hell. I believed that Heaven was a place of eternal bliss and relationship with God while Hell was a place of everlasting torment in fire and agony.
The interesting thing that I found though was that the idea of an eternal fire laden Hell is actually one that has evolved over time within Judeo/Christian thought. The earlier Jewish writings [earlier Old Testament books] do not speak of a life after death. They might speak of a paradise type of situation for the righteous but they leave the impression that the evil and wicked will just be eternally snuffed out, so to speak. It only speaks of them going "to the grave" and not to Hell.
I also found through my studies that the mental images that Jesus used in his teaching didn't really seem in line with the idea of eternal torment either. The picture that Jesus painted was based on a real place outside of Jerusalem called "The Valley of Hinnom" which was a dump where the garbage was burned; the fire burned continually because they were always throwing stuff in there, but the individual items eventually were fully consumed. This led me to the belief in what is called
annihilationism. It basically means that those who will experience "Hell" will experience a time of suffering but then ultimately their spirit/soul/mind will cease to exist.
So coming back around to the original question- are we originally destined for Heaven or Hell?
For those who take the view that there is an "age of accountability" they believe that if someone dies without reaching that point that they get a free ride. If they do reach that age and choose to reject God/Jesus then they get the hot plate.
For those who take the view that God has already decided then if you're not part of the club then you're out of luck; no matter how hard you try or how unfair it might seem.
My only challenge for those who believe in Hell according to the Bible is to make sure that their idea of it matches up with the historical, literary, and allegorical presentations presented in the Bible rather than the concepts of firey pits that more reflect Greek and Roman mythology.