Big Show venue has always been the hardest for me. It doesn't even appear in the drop down box UNTIL you actually have a BSV placed somewhere in your town. I have the KP one, (which isn't any better or prettier than Binder Clips (stadium for athletics AND BSV), or the other one which you can save to library from SS. Once you've put a BSV anywhere in your town, (I stick a big lot in the middle of nowhere and plop one down just so the drop down box shows the listing), you can build one using the biggest stage and that black crowd floor. You also have to use the other floor markers though if you actually want people to come to the show.
The Big Park and Coffeehouse are both easy. As Tom said, just add a stage to a diner and voila, and just add a stage to an already existing park. Private venue is any building with the appropriate stage in it that you designate as private venue. Ausette made one for UC and it works very well. Same goes for Live Show, add a stage to pretty much any building, (don't forget the floor markers from buydebug!), and you've got it. Actually, it doesn't even really have to be a "building". I turned the pier in UC into a Live Show venue by adding a stage and seating.