I believe, your question depends heavily on what kind of developer you are talking to. Since most people here are Web Devs and people already asked what tools everyone uses quite a few times, I will try to answer the question from the perspective of a low-level game developer using Rust this time.
First, Rust still does not have any elaborate IDE. There are only a bunch of plugins, none of which I would call complete by far. So, at the moment, I go with CLion+Rust plugin as IDE and it works out (kinda). In addition to a full blown IDE, I really like to keep a normal text editor around. I love the light-weight Kate for that kinda job!
Since I want to use Servo for the GUI part, I also need Web Dev tools, like WebStorm and Chrome's developer tools . But I don't want to go deeper on Web Development here :)
When having a tool to write code, you also need a tool for versioning it. Personally, I prefer gitless+GitHub, since it offers me great freedom and I am familiar with it. I also use GitHub for project management and issue tracking.
The third thing I constantly need is a browser (I prefer Chrome and Firefox) and Alphabet's Google in order to look up documentation.
Last but not least, I use Cargo a lot for building and Crate management, though I do not use a lot of crates (there are not many and most are still in very early development).
In addition to the programming part, I also need Audacity for sound editing, Blender for 3D asset editing and GIMP for 2D asset editing. I am not a sound or 2D/3D artist, but a developer. However, even then I sometimes need to take a look at files and maybe even make a little tweak myself. So for me, they really are necessary.