Congrats on your new ereader!
1. When you connect the reader via USB, it will show up like a disk drive in My Computer on a PC (Finder window on a Mac). Double-click through the folders to database > media > books and you will find the books there. You can drag-and-drop or copy the files from the reader to your hard-drive to make a backup copy. But, like Pinecone said, if they have DRM, then they won't work when you authorize the device to your Adobe & Sony account info.
2. I like the Sony Reader Library. I use it as my primary software to transfer books. Don't be scared off by other opinions. The performance varies depending on your computer hardware and operating system. It doesn't hurt to at least try it out and see how it works for you. Calibre is good too and at least you know it's another option if you don't like the Reader Library.
3. Along the lines of what Pinecone said, also don't forget to "eject" the reader before you disconnect the USB cable. Also I recommend reading through the sticky threads at the top of the Sony Reader forum. For example the sticky threads help you with other sources for ebooks besides the Sony Store and how to use library books. I would read through the PRS-600 User's Guide too. You can find it at this link.
http://esupport.sony.com/US/perl/mod....pl?mdl=PRS600
4. Pinecone already answered this one.
Good luck and happy reading! I hope you enjoy your new purchase.