This is a tricky question, starting from the difficulty to define religion and spirituality and turning even harder since the people we talk about were very different from us. There are traits that can be found in how our psyche works as well as in what archaeology has brought us. I think one way of gaining knowledge of primeval religion is by looking at how children view the world.
1) The natural tendency to see consciousness where there is none
This can be simplified to the idea of seeing tigers in the bush even when there's no one there. Many forces of nature seem similarly wilful when the person is unaware of the laws behind the phenomena. Thus, the sky decides to rain and the wind decides to blow.
2) Magical belief and false causality
I think all of us are aware of how hard it can be not to attribute your luck or misfortune to something you've done. Our brain naturally "clicks" into a thinking where sheer chance doesn't exist. It's been shown that all animals fall for this, like when pigeons develop strict rituals to gain treats from a machine that in reality works purely on random.
3) Added value to things we rely on or put much effort into
Our brain is wired to stick to something we've put much time into. Likewise we put less value on things we aren't able to gain (see cognitive dissonance for more). This means we naturally elevate some things to a special position. For hunter-gatherers these could be family, prey, fertility and safety. This could explain the Venus figures and cave paintings as well as the wide range of imagery picturing human figures in various roles.
4) Group thinking
Humans are group animals. We thrive in tribes and form hierarchies based on who we value in the group. One very big part of religion are the ceremonies that are observed in order to preserve the order. For us modern day people this could be important family holidays like Thanksgiving and Christmas. For ancient communities it was even more important for all members to know their place and feel that the society was worth thrusting.
I might have forgot something. That's my thoughts for now anyway. Thanks for the interesting thread.