In my never-ending pursuit of information on the early years of American comic strips (hey, everyone has to have a hobby) I’d really love to get hold of documentation about the lawsuits between newspaper magnates Hearst and Pulitzer from the turn of the century (the next to last one, that is). They were always firing off lawsuits back and forth over copyright violations as each stole the other’s cartoonists and comic strips. I don’t know how many of these suits actually made it into court but I’m guessing there must be some record of the filed actions even if they patched things up out of court. I don’t have a clue how to even start a search for such records. Suggestions?
I’d look thru the archives of the old newspapers during the time the lawsuits were active. Surely there would be stories saying something like " Hearst filed suit against Pulitzer in such-and-such court today". Then you go to the legal archives of that particular court and look them up.
<<Then you go to the legal archives of that particular court and look them up.>>
And how do I do that? I think it’s safe to assume that all these lawsuits were filed in NYC in the state of New York.
If the suits were filed in federal court, then there may be some record at the National Archives. Their website is archives.gov.
Well, let’s say the document was filed in the chancery court. You would find that out from the newspaper story. You waltz up in there and say to the clerk " I’d like some help finding this lawsuit which was filed in chancery court in February of 1903" or whatever. Then they help you find it. 
Have you thought of looking for biographies of these men? Perhaps some more info could be found there.
You’ll want to start with the New York State Archives.
It appears as if they have a fairly good collection of court records and the like, and some finding aids can be found here. If it’s civil suit records you’re looking for, you may have to focus on local government holdings.