After testing, here are the results: (Please consider that I only used each editor for minutes ! This was just to understand the differences for the setup and quick use)
Vim is complicated to setup (but once you get used to it there are no issues and it gets easier as you progress). It was REALLY slow to open up the first time and then was fast enough (under 5 seconds is OK, I don't close vim usually, I have keybinds for everything that I need). You need the syntastic plugin as well as the youcompleteme for it to be usable (IMO). Then you can add vim-snippets and all of the plugins that you want to get the experience YOU want (this is what I love about VIM). Youcompleteme can be slow at times. EDIT: Correction, Youcompleteme is really fast, shellinabox is not though, and this is what I'm using most of the time right now. What is slow is Syntastic, but I found the excellent ale plugin that is async in Vim8 and Neovim (I didn't have Vim8 or Neovim on my server). I highly recommend Ale from now on for programming on Vim unless you need a language that it doesn't support.
Sublime Text 3 is fast. it only needs Rust Enhanced and RustAutoComplete to be fully usable. It's easy to setup (only RustAutoComplete needs to know where your sources are and you're ready to go). It seems like it's the fastest (using racer).
Visual Studio Code was a first for me. I tried it with racer and it's slow to update... Like a minute or two slow... But with RLS ON, oh boy, it's not the same experience at all! real time errors are great! Definitely the fastest BUT you have to have the nightly toolchain installed. Only one extension is needed, Rust and you can choose slow mode (legacy) or RLS (really, realy fast). Interface is different but I like it ! AND I'm not done! It works on Windows ! You only need to install RLS with the instructions from their git.
Atom is slow and I just don't like the interface. I don't know why. It can be useful for web devs but for rust it takes over a minute to update. Too slow. Maybe the languageserver-rust plugin would actually make it usable, but I'm biased here. I don't like this one for reasons (I don't even know myself). It still seems like a great editor. Your choice on this one, test it.
IntelliJ Rust doesn't seem to work well enough for me. The simple test of writing:
let string: String = "hello";
Didn't return an error without me having to build the thing. Not ideal. When it will be more developed it would be a no brainer for anyone that is used to the Jetbrains tools. Didn't pass for me.
Conclusion:
After a quick test, Sublime Text and Visual Studio Code seem to be the best choices. Sublime is fast, even with racer. I've also seen some discussion on RLS for Sublime and this: LSP. Maybe this is worth giving a try ?
Visual Studio Code is ONLY usable with RLS and this is why it's not my first choice. To use RLS you need to have the nightly toolchain installed and I think it is unlikely that you are using the same toolchain to compile and run your projects.
My own preference goes to VIM because I'm getting used to it and really like how it works. Plus I mostly work directly on my server right now because I am making a web server backend for my website. So VIM seemed logical for me.
Don't take this too seriously, I did it in one night. Just to try everything out. @CleanCut I think you should try Sublime Text 3 and Visual Studio Code yourself.