How does random change (mutation) in the genome add information to a genome to create progressively more complicated organisms?
It Doesn't. Evolution is Non-Random. James A Shapiro says "Genome change arises as a consequence of natural genetic engineering, not from accidents" (from James A Shapiro_2009_Revisiting_Central_Dogma).
A multi-national team of biologists has concluded that developmental evolution is deterministic and orderly, rather than random, based on a study of different species of roundworms. Another molecular biologist from Princeton, David Stern, says that evolution is predictable because mutations tend to occur at hot spots in regulatory genes. Even Richard Dawkins, an atheist and Darwin's most vocal defender, says that evolution is non-random, referring to Natural Selection as a "non-random mechanism" in a Salon interview.
Lack of Gradualism in the Fossil Record
Another flaw in Charles Darwin's original theory is the geological record which did not support Darwin's prediction of gradual transition between life forms. To explain this lack of gradualism in the fossil record, Stephen Jay Gould invented a new theory called punctuated equilibrium. Rather than finding gradual transition forms in the fossil record, paleontology has found sudden bursts in appearances of life forms, as well as long periods of stasis or little change in the fossil record.
One example of this is the Cambrian Explosion about 555 million years ago when 40 body plans appeared over a ten-million-year period with no preceding transition forms.
These findings had to somehow be incorporated into Darwin's theory, adding to the complexity of the theory. Thus, the invention of "punctuated equilibrium".
Cornelius Hunter, author of Science's Blind Spot, regards punctuated equilibrium as a form of "post-hoc" reasoning to be analogous with the Epicycles of the pre-Newtonian Geo-Centrists who claimed the earth was the center of the universe. Galileo's telescope disproved Geocentrism. Left Image Courtesy of Discovery Magazine.
How does the first living cell arise spontaneously to get evolution started?
The spontaneous origin of life is called abiogenesis, and neo-Darwinism does not even address this issue.
Science doesn't have a clue as to how the first living organisms originated. Stephen Meyers refines this question further as the question of the origin of "biological information" encoded in the DNA of the first cell. We have no scientific explanation for the appearance of this first information.Central Dogma Refuted
The central dogma says that genetic information follows in only one direction from the genome to the protein. The discoveries of reverse transcriptase and retroviruses showed reversed flow of information back to the genome from the environment. A new concept called the "fluid genome" replaced the central dogma. This has profound implications for evolutionary theory.
If you plan to study the topic of evolution, a good place to start is Gert Korthof's web site, which exhaustively reviews dozens of new books on evolution, intelligent design, evo devo, molecular biology, etc.
Mainstream Scientists Who Dissent
As of April 2014, here is a list of scientists who dissent from Darwinsim. They all signed this statement: