Buddhism derives from a Hindu background, and the basic philosophy is the four noble truths:
1. Life means suffering.
2. The origin of suffering is attachment.
3. The cessation of suffering is attainable.
4. The path to the cessation of suffering is removing attachment.
Taoism has its origins in China. There are two forms, the later form adopted the original philosophy, but with a prime objective of achieving eternal life (with potions or breathing practices etc). Ironically this was completely contrary to the original philosophy of taoism.
Taosim philosophy is based predominantly on the writings of Chuang Tzu (350 BC). Although the other key texts are the book of lieh tzu, and the tao de ching (by Lao Tzu).
Chuang Tzu was effectively an contra-rationilst. An important point he made is "how can we use what is limited (the brain) to understand what is unlimited (the universe)." Its for this reason that Taoism often seems contradictory or strange, because the purpose isn't necessarily to 'prove' what we know, but to understand that we cannot conceive of reality intellectually*. Talking about knowledge or understanding he would say...
Great knowledge grasps the whole; small knowledge, only a part. Great speech gets hold of the universal; small speech picks at particulars.
For whether our mind is fenced in by sleep or released with the body in waking hours, it is buffeted by daily mental ferment—indecision, bafflement, deviousness, fretting anxiety, and quaking terror. At one moment the mind flings out like a javelin, to arbitrate right and wrong. At another, it is a solemn holder of rights, guarded and secure.
Then, with the blight of life’s autumn and winter, comes gradual decay, a passing away, like water draining out, never to return. Finally comes the blockage, when all is choked up like an old drain—the failing mind that shall not see light again.
Joy and anger, sorrow and happiness, caution and remorse, come upon us by turns, with ever changing mood. They come like music from hollowness, like mushrooms from damp. Day and night they alternate within us, but we cannot tell whence they spring. How can we hope in the spur of the moment to lay our finger upon their true cause?
Without these emotions I would not be. Without me, they would not exist. So far we can go. But we do not know what brings these emotions into play. It would seem to be something in charge, but the clue to its existence is wanting. That something is actively in charge is credible enough, though we cannot see its form. Perhaps it has functions without form.
There are also relativisitic aspects in Taoism i.e. "greatness" and "lowliness" just being matters of perspective.
Zen Buddhism is Buddhism that was influenced by Taoism, and became prominant in Japan (though the origin is China). It is like a formalised version of Taoism, with religious practise and often 'koans' which are often weird or make no sense. The idea being to break out of the normal thought processes and realise things beyond simple conceptualisation of the world.
*(interestingly Aristotle said the same thing)
P.S. I absolutely recommend reading "The Way of Chaung Tzu" by Thomas Merton. Chuang Tzu changed my whole perspective on life.