I did not know you could do that (look a the county website). Do you know if every county makes that information available through a website?
I believe most if not all counties do list cases on their websites which you can search. It ought to contain the case number, parties, lawyers if any, petitions, motions or judgments filed and the next scheduled hearing. Generally you can't read the full texts online, you'd have to go to the records office and ask to view them.