How are the gene sequences of individual sperm and egg cells "randomized"?
Meiosis is the type of cell division responsible for the diversification of genetic material among egg and sperm cells. The diversity comes primarily from crossing over (Prophase I) and the cell divisions (Telophase I &amp; II) later on in the process.


  Meiosis begins with one diploid cell containing two copies of each
  chromosome—one from the organism's mother and one from its father—and
  produces four haploid cells containing one copy of each chromosome.
  Each of the resulting chromosomes in the gamete cells is a unique
  mixture of maternal and paternal DNA, ensuring that offspring are
  genetically distinct from either parent. This gives rise to genetic
  diversity in sexually reproducing populations, which provides the
  variation of physical and behavioural attributes (phenotypes) upon
  which natural selection acts.


Further detail of the precise phases can be found in the Wikipeida article.


[picture from wikipedia]