How does water and nutrients travel up a plant's stem going against the force of gravity and grow themselves for many feet?

It happens through capillary action, which is an effect connected to surface tension in water.

If you look at the water in a glass drinking-cup you will see that the water curls up at the edges to form a meniscus. That is caused by surface tension. In a narrow tube the effect is more marked, and in a very narrow tube - a capillary - the water is driven up continuously. That is how plant life is possible, and of course it led on to animal life and eventually to human life.