I think learning C first, let's you appreciate Rust a lot more and also helps a lot in understanding it. C is a prettg raw and dangerous language, but it is also close enough to the machine, while not being a system specific assembler. So it can teach you about memory layouts and about the dangers of manual memory management and data races.
While Rust probably could teach you that, I think it is more likely, that you get annoyed, why you should have to deal with a stupid borrow checker. But when you already know, what is dangerous, your probably glad to have it. Also I think that C is just closer to how CPUs operate, so it is good to get a basic feel of low level programming.