What is the best transcoding type for my current file setup? (more in desc.)

[User 14372]

I have discovered that every single device that I use cannot play back any of my movies without heavy transcoding. My media library is currently just DVDs that have been ripped by MakeMKV. I have also included all available audio tracks (with 5.1 and stereo overlap), and all subtitle options. I am doing this in order to make sure that everything is being copied, just in case, and I cannot afford to lose any of it. I have discovered that whenever I play movies on other devices, the video becomes blocky. Jellyfin's info panel tells me that the video is in MPEG2VIDEO Main video, MKV container, and AC3 audio. I want to change the video type/format now, to make my movies play without transcoding. I use Linux, so I will be using Handbrake (since it's a GUI, and I don't want to use any terminal programs like ffmpeg just yet). What would be the best way to keep all of the movie's data (video, sound (all types & languages), subtitles (all languages)), but make it be as easy to play on as many devices as possible? I am fine with keeping the MKV container, as long as Jellyfin doesn't make my movies blocky like it does when transcoding.

(Jellyfin Desktop and VLC play the original MKV file as from the DVD source, Jellyfin transcodes it in such a way so that it becomes blocky, and the colors become different(?))
(sorry for the wall of text, I haven't done transcoding ever, and I'm worried that I'll have to rip all of my movies again if I do something wrong)

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Just to clarify, DVD plays good, VLC plays the MKV good, Jellyfin Desktop plays the movie good, other Jellyfin clients transcode basically everything and create a blocky mess (maybe this is something to do with the movies being 5.5GB for 1h45m of video).