There are two support documents you should peruse. Both will help with running Plex on a Synology NAS:
Installation & Setup of Plex Media Server on Synology
Synology FAQ - Questions, Answers, and a few How-To’s
Installing / uninstalling Plex Media Server on your Synology NAS will not affect your media files (movies, shows, etc). The installer installs/removes the PMS software. It does not touch your media.
Login to your NAS via Synology DSM (Synology’s web interface). Using File Station, do you see your movies, shows, etc?
Did you install your media in the Plex shared folder? If so, you will have to move it. That folder is reserved for use by Plex Media Server. See the Installation and Setup guide linked above for details on media organization.
Please pull the Plex Media Server log files and post them to the thread. Somebody will be able to take a look and assist.
If PMS is running, login via Plex Web and go to Settings -> Troubleshooting and click on “Download Logs.” Attach the entire ZIP file to the thread. Drag the file into the window and it will attach at the cursor location.
If PMS is not running, then login to your Synology and launch File Station. Navigate to Plex -> Library -> Application Support -> Plex Media Server. Right click on the Logs directory and choose “Compress to Logs.zip.” When the compression has finished, download the ZIP file (right click -> download) and attach the ZIP file to the thread.
The Plex shared folder may be hidden. If so, you need to change the permissions to make it visible. See Synology FAQ #1 for instructions.