"How can one believe in evolution considering the staggering complexity of DNA and other parts of cells, and considering that , relying on chance alone, it seems impossible that life could randomly arise, even given trillions or sextillions of years. (Billions wouldn't be enough)?"
This is a very ticklish question, and when it comes to me, I always like to check to see if the person asking is seriously interested in the question or is more interested in arguing with the science teacher. Because I don't know you and can't do this check, I am going to assume that you are interested in a serious answer. As a rule I don't engage in 'evolution' vs. 'intelligent design' type debates.
So what are the scientists thinking?
When we think about random processes, one is tempted to imagine that DNA could have arisen by taking a big, very big, bowl of assorted amino acids, and stirring it for a few billion years. This doesn't come close to describing the complexity of the evolutionary process which is full of feedback loops, both positive and negative.
Here is a very simplified analogy: There is an old question about if we gave a million monkeys a million typewriters (and the inclination to type) for a million years, would they reproduce the works of Shakespeare, or the King James Bible? The answer is no, it would be very unlikely either work would appear out of that random process. On the other hand, imagine you gave just a hundred monkeys the keyboards to a hundred networked computers (and the inclination, again, to type) and every time a monkey added a correct symbol to the text, all the neighboring computers reset themselves to the completed correct code to that point, and when monkeys did not contribute correct symbols, the computers would not. It would hardly take a hundred years to type out the text of the King James Version of the Bible.
This example might be taken to suggest I am implying a design. The computers the monkeys are using would have to already know the text it was looking for. But evolution takes place in a lot of different places with different environments and climates and minerals and other resources and on top of this there is constant change. So evolution is based on an idea that there is no one right answer for everywhere and always. (This is why geraniums, oak trees, octopuses, cockroaches, bacteria, and, so far, human beings have all been able to thrive while being completely different kinds of organisms.) Every time an organism chances on a 'right' answer, or gets lucky, it gains an advantage over its cousins for the time being. When it chances on a 'wrong' answer, it loses some advantage.
Whether a particular modification is a 'right' answer or a 'wrong' answer for one individual can depend on which side of the ridge it sits on, which way the wind or current is flowing, the salinity of the water, the acidity of the rain, etc. Far from being a random operation, natural selection drives change and adaptation, but it doesn't drive it in a particular direction. There is no ultimate object to evolution, and a clam, a hummingbird, an amoeba, and again, a human being are all equally valid solutions to different problems. Dinosaurs had all the right answers for millions of years and then suddenly the questions changed, and their right answers turned into wrong ones for the new problems.
I am not an expert on natural selection or evolution. If you really want to see the thinking of a highly qualified scientist, I recommend any of the late paleontologist Stephen Jay Gould's books. His essays can seem wordy sometimes, but he tried to be very clear about the concepts he discusses. You can find his books in a library. I have enjoyed The Panda's Thumb, The Mismeasure of Man, and Ever Since Darwin, each written for general readers.
I hope I have helped in part to answer your broadest question, and I hope you haven't noticed that I have tried to slip past your narrowest one. The actual origin of life on Earth is not well understood, and it may have actually taken literally billions of years and just the right sequence of events and physical circumstances. Once life got started, though, given the variety of environments and the machinations of variation and natural selection, the diversity we see today was just a matter of time.
