What exactly did the modernist writers want to achieve through their writing?

This is a broad question, but a good one.  There are
exceptions, but in general, modernist writers' goals are focused
on:

• the replacement of religious
  certainties, moral absolutes, essentialism, and determinism with
  skepticism, doubt, and relativism


• a feeling
  of alienation, angst, and "nausea" about materialism, mainstream
  institutions (education, work, and religion), role of the individual in society, and the
  accepted, status quo standards of
  society


• existence as problematic
  (i.e., there are no certainties or absolutes about the meaning of life,
  the existence of God, the traditional pathways to
  happiness)

Remember, the modernist era began
with incredible hope and improvements in technology (the camera, internal medicine,
airplane, mass production, industrialization), but it was plagued with nationalism,
greed and poverty, and cruelty--all culminating in the worst 30 years in human history
(1914-1945), which saw two world wars, a great depression, pogroms and holocausts, and
despotism and corruption.  The sum total was "engineered pain": we had found a way to
kill and torture more efficiently that ever before.  So, you can imagine that wholesale
changes in beliefs about God, country, and the individual were taking place--most of
which were negative and pessimistic, but justified.