The first step in finishing our basement was to replace the drains underneath the basement floor. We have not had any problems with the drains, but we figured it would be smart to replace them now and avoid a mess in the future. I broke up the concrete floor on the half of the basement that has the drains, and then dug down in the dirt to reveal the old drains. We quickly learned that the drain line from the house to the street (i.e. under the front lawn) had been replaced by the previous owner (bonus!) and that we had no signs of tree roots in the drains under the house (double bonus!).
We hired a plumber that I had worked with previously to replace the drain lines for us. I probably could have done this work myself, but we were in a crunch for time and I did not want to mess something up and realize this after it was covered with new concrete. He basically just replaced the old clay drains with PVC, as no major changes in layout/slope were required. He also roughed in the basement toilet and shower, as well as a stack at the back of the house that we will tie into for the future first floor washroom.
At this point, we just piled all of the concrete and pipes into a huge pile in the corner. Breaking up the concrete was no big deal because it was thin and mostly mortar. Removing the soil was much more difficult, as I dug it into buckets, carried it up the stairs, and piled it underneath our front porch (we had some grading issues under there that we wanted to solve).