Welcome to JavaRanch.
First off, be aware that using a shared folder limits the use of the applet to everybody who can access that folder - say, everybody on the same company network. Nobody outside the firewall will be able to use it. Just something to keep in mind.
Something else to keep in mind is how to organize the files in the shared directory: How do they get unique names? How do you make sure that files uploaded by different users don't have the same names and so overwrite each other? How can a user go back to view all the files she has uploaded? How does the code know which user has uploaded which file?
JFileChooser can only select a single file. If you want to upload multiple files you'll have to invoke it multiple times and upload each selected file individually.
As to transferring control to a different applet, the upload applet needs to remember all names/paths of the files. Then it can either use the AppletContext's showDocument method to switch control to a different page that contains the viewer applet (and pass all file names in the URL), or it can first save the file names to the server (by doing an HTTP POST to a
servlet or similar), and then call showDocument to a server URL that knows about the previously POSTed file names, and which can then create the page with the viewer applet and add all the filenames as <param> tags to the <applet> tag.
The former approach (passing filenames in the URL) limits you to just a few files, since URL are generally restricted to 2000 characters or so.