From what I learned in my history class it seems they didn't really do much. Prussia and the others just came in and said, "this is ours."
Is there any record of what the people of Poland-Lithuanian did in response during that period? Did the people riot, did they rebel, did they form militias to defend their home?
Well there was the rebellion of 1794. Sometimes it's called the Kosciuska uprising. People did form local militias during those times, had a officer in villages equivalent to a sheriff, only answered to polish - lithuanian government. It was after the second partition(1793) that people got real serious about the occupation. Before that there were only things like Constitution of May(1791) and confederacies but all were overwhelmed by Russian, Austrian or Prussian governments. The third partition(1795) happened so quickly because of the rebellion.
Afterwards, patriots of the commonwealth really hoped that other european countries would free them. There were a lot of former soldiers and nobles from Poland - Lithuania who joined Napoleons France, had separate units there(ex. Polish legion) and believed that Napoleon will eventually conquer Russia and free the Commonwealth.
Later on there were uprisings in 1831 and 1863 - 1864, but I assume you were asking about actions taken immediately after or during the partitions.