Players and Platforms
The beautiful thing about Plex is that the developers take pretty much any platform seriously. This means you'll be able to play Plex-content on about every device imaginable. How well it works depends, but there's a pretty good chance a(n official) client is already available.
I'll describe some of the most-used platforms and recommended clients and players here.
Platforms:
This section is about platforms people will probably already have available. They're just recommendations/best practices I discovered through my own experiences. These are not necessarily my most recommended platforms/players, they're just the most practical without any investment.
Microsoft Windows
Probably the first choice for most people after getting into the Plex-server, although not the best choice in my opinion. I think most people try Windows first because they never heard of Plex and don't know about its great availability.
Windows has two options for playing Plex content: the browser and a dedicated app. If you're set on using Windows, I'd always recommend the dedicated app. The app is way better as it supports way more codecs ("filetypes" for video files) and just plain feels better to use. The dedicated app can be found here or through the Microsoft Store. Make sure you download Plex for Windows, not Plex Media Server (PMS), unless you want to set up your own mediaserver.
Android/iOS
I'm talking about all mobile devices running Android or iOS (yeah, I know Apple makes a difference between iOS and iPadOS, but let's be honest, it's all the same). There is one big limitation to the mobile Plex apps: they're "not free".
Plex has always been, and will always be free for the "big screen" (TV's, mediaplayers, PC etc.). However, for people who do not subscribe to Plex Pass or have a lifetime Plex Pass they do limit the functionality of the mobile app (stop playback of content after one minute). I believe they do this to cover the cost for publishing apps in the Google Play/App Store. There's three ways of solving this: pay a one-time-fee of about €7, subscribe to/buy Plex Pass (€5 monthly/€40 yearly/€120 lifetime) or just use the mobile app to cast content to another (big screen) device. I'd personally recommend just paying the one-time-fee or buying a lifetime Plex Pass. To me it's more than worth it and it makes sure the developers get paid and have reason to continue this great project.
The Android-app is available here and the iOS-app is available here.
Smart TV
Most people own a TV with some smart capabilities nowadays. Most Smart TV manufacturers use their own OS (e.g. LG with WebOS, Samsung with Tizen etc.) with their own respective app store. I can't very well describe the ins and outs of every platform as I can't reasonably own/test all of them.
I have verified the Plex-app to be available for LG and Samsung TVs. The respective apps work decently, albeit a little slower/less stable than dedicated media players. TVs tend to have fewer resources/fewer development time spent on them as they have fewer users as well.
If you verified any other Smart TV platform to be working, please leave a comment down below.
Players:
Chromecast with Google TV/Google Streamer
Nvidia SHIELD TV Pro
Apple TV
Game Console
Microsoft Windows HTPC
Chromecast V2/V3