I'd echo the sentiment that individual's will differ on this topic. The most predictable awakenings will be a person's awakenings to a shift in desire(s). When we are young we want certain things and growing older we discover than our earlier desires no longer hold the same place in our values. We want new or different things.
At one point we might value freedom and adventure above all other things. Later we might realize, sometimes suddenly, that we really want to settle down and start a family.
As the people stated already, all people will differ in their opinions. In my experience, many people experience the fact that material things do not create for happiness. Kids learn this at times when their toys no longer make them happy. Adult learn this as they realize that they want more and more. Hence, people come to the realization that there must be something more apart from materialism.
At some point in everyone's life, and probably at more than one time, a person will experience a "loss of innocence," and that is probably going to be the seed of some of the more important awakenings one has. Those losses reshape a person's understanding of himself and/or the world around him. Those losses can take many shapes, but loss of loved ones, betrayal by friends, significant failures are common losses.
When we are young, we learn about the world spiritually and philosophically through the adults around us. We get an education at school, church and home. As we grow, we begin our spiritual and philosophical awakening by questioning what we have been taught. This usually begins in adolescence. Then we develop our own spiritual and philosophical beliefs, but they don’t stay static. We grow and change as we age, and as we have experiences.
I agree with the previous post in that awakenings will vary from person to person. Many people I know have seemingly become more religious and conservative in their worldview as they have become older, taken on more responsibilities, and experienced more of the world. I find, generally, that I have gone in the opposite direction. I think some people philosophically come to understand the world as being more complex as they get older, and are less inclined to seek simple solutions or to regard issues as "black and white." But again, I think many people become more entrenched in their own value systems. The point is that whatever "awakenings" you experience will be based on your own life experiences. Development is, I think, a subjective process.
This will differ from person to person. Some people will become more spiritual as they age, possibly because they become more and more aware of their own mortality. Others might become less spiritual as they come to feel that their experience of the world reveals to them that spirituality is pointless.
Philosophically, you would hope that most people mature as they grow older. You would hope that they come to value the material things in life less and come to value things like the importance of family and the importance of giving less attention to their own wants as compared to those of others.