It's a small language, it's everywhere, and it's going to build character.
Rust is still undergoing active development, and solves a different problem space than C by virtue of what it adds. It's a good language, mind you, but it doesn't expose you to the warts of hardware like C will (by design!). It won't let you deal with null or dangling pointers. It will abstract away data types with let. It will treat memory safely (which it really isn't). It's a great language that will hopefully replace C one day, but that doesn't mean you should start there.
Learning and debugging C will help you understand how processors work (especially if you start with something simple like an MSP430 devkit). You can look at the code and the generated assembly and see pretty much how the compiler came up with what it did. If you're developing on a Unix descendant, you'll be dealing with things at a level the operating system intends.
You'll get to look at a huge corpus of code already out there and see what works and what doesn't, and if it strikes you you'll be able to modify it (say, making mods for Half-Life or Quake 3 or whatever) and see how things work.
Start with C!
EDIT:
Later, you'll see why C needed replacing. You'll never appreciate Java fully, for example, without dealing with all the stupid C++ allows.
Comedy option--learn Erlang or Elixir--you'll be spoiled for the rest of your career once you see how annoying things can be elsewhere.