It's good to understand the underlying systems concepts in general (such as "what is the difference between heap allocation and stack allocation?"), but you don't necessarily need to know C to grok these concepts. However, C is still a good language to know, or at least be familiar with.
As to Rust, there are still a few essential constructs missing from the language, so if you're looking for a gentle introduction you'd probably be better off waiting for 0.3 (scheduled for release in "about a month") or 0.4 (probably due to be released in three months or so).
As to Rust, there are still a few essential constructs missing from the language, so if you're looking for a gentle introduction you'd probably be better off waiting for 0.3 (scheduled for release in "about a month") or 0.4 (probably due to be released in three months or so).