Jellyfin or Plex, for sure. Plex is a fork of Kodi, and has a premium service you can buy. Jellyfin is the open source alternative. The program will search a specific folder for media, link it to known shows/movies, download the metadata/posters/trailers/etc and make it available for streaming. The server will allow for direct play when possible, or will transcode the media into a different format if the streaming device isn’t capable of playing the file directly. This last part is where many people run into issues, as they’ll try to run it on something small like a raspberry pi, then get stuttering when trying to watch 4K streams, because the server isn’t powerful enough to transcode the media fast enough to keep up.

For remote streaming, you’ll probably want a VPN; ISPs tend to get squirmy if they know you’re hosting a server for copyrighted content. It will also require port forwarding to be able to connect to anything outside of your local network. So your use case will definitely require port forwarding; When selecting a VPN, be sure to pick one that supports that.

As for setting up the server, it’s basically a matter of placing the media into specific folders, then telling the program how those files are organized. Plex will pull file data directly from the file names, so a program like FileBot to automatically pair files to known databases and rename them to Plex’s naming scheme will be a fantastic tool. It makes file management much easier, because you don’t need to worry about mis-matched shows. Then once the server figures out which files are which, it will automatically deal with the metadata, image downloads, etc…

Then for streaming, it’s a matter of connecting with your Plex/Jellyfin app, which will find your server and initiate the stream. With PlexPass, you can even do things like download files locally, then your device will simply update those files whenever it reconnects to your server. I use this on my iPad to watch stuff at work, then it automatically deletes the watched episodes and downloads the new ones when I get home every day. But not everyone wants to burn that much storage space for videos, so streaming remotely is also possible.