My first Lisp learning experience was with Scheme, I've never touched Common Lisp (felt it was too complex), and am now starting on Clojure.
I used Dorai Sitaram's "Teach Yourself Scheme in Fixnum Days" to learn Scheme and got fairly far though I never really found myself wanting to use Scheme in real projects.
Clojure, because it purportedly gives nice, clean access to the huge universe of J2SE/J2EE libraries, on the other hand, encourages me to relearn this Lisp dialect because it may finally be of practical use.
As for which one to start with, I would say Scheme is simpler and so might be more appropriate to start with. On the other hand, if you have good Java and Python knowledge, you might not mind diving straight into Clojure because, unlike Scheme, it contains elements of these other two languages (e.g. data structures reminiscent of Python and JVM/Java API centric tutorials) and the familiar terrain might help.
Since I did come from all three (Java, Python, Scheme), I find myself in a good position to appreciate just what Clojure brings to the table that is different from Scheme. I'm no experienced Schemer, but I'd say that if you immediately start with Clojure, you will still get the general Lisp experience, so you definitely won't be missing that by forgoing Scheme.