If you want to play, just play with the toy that looks the most fun. Don't plan for the best route, take the most interesting one. There will be enough occasions in your coding life to be serious and not be given a choice...

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).

I'm not really sure that Rust is quite ready for much usage yet, always worth a try, but some simple things are still pretty hard.

Just learn C.