You want to look for a balance between simplicity and cleanliness, attractive features, and a platform that will allow you to write interesting and useful software (to yourself) as well as serve as a learning tool. (This last will keep you going and learning for a lot longer.) Here are some possibilities:
Scheme. Probably the cleanest of all dialects. This is no doubt why The Little Schemer was translated from LISP into Scheme. The fifth Scheme standard specification, R5RS, is a wonderful education in and of itself; it may be the nicest language and library specification I've ever read, as well as the shortest that's reasonably comprehensive. The PLT Scheme (now Racket) platform includes a fairly decent interpreter and compiler, is good for scripting, and also has some visual tools that make it excellent for learning.
Common Lisp. Probably the most portable and comprehensive variant, this is most likely what you want if you want to be writing things such as commercial software. The standard defines extensive libraries, and many more are available beyond that, it has CLOS, which will probably teach you more about OO than any OO language could, and some of the compilers are very good. Disadvantages include some warts that Scheme doesn't have (such as having a separate namespace for variables that refer to functions), not being as clean and simple (as is the case with anything that has had to have the extensions and make the compromises necessary for large applications in the real world), not having hygienic macros, and emphasizing recursion much less than Scheme.
Clojure. This runs on the JVM, which may give it a leg up right there for Java developers. It's got a few warts (e.g., you must explicitly ask for tail call optimization, though this may change one day if TCO is added to the JVM). The macros, while not hygienic, do have some features to help you avoid variable capture, so you can capture variables if you really want to, while running less risk of accidentally doing so than in CL. You've got easy access to all the Java libraries; that's probably a good thing for "real world" code, and fairly pointless in terms of learning. It's got a set of libraries for persistent data structures and support for STM, which make it very interesting from a concurrent point of view; this makes it probably your best bet if you're interested in learning more about new methods of dealing with concurrent and parallel programming. It appears as if Clojure is as usable for large, production applications as Java, in the sense that it's going to have the ability to do the "ugly stuff" you do in production apps that you'd rather not do and don't do when you're learning.
Emacs Lisp. In terms of a LISP, this is not one of the better examples out there. One of its biggest faults is dynamic scoping, but there are many others. However, if you're an Emacs user, this may be the most powerful tool you can learn to improve your use of the editor. How much you'd really learn from learning Emacs Lisp, beyond how to extend Emacs, is for me an open question however; I don't know how often interesting techniques such as high-order functions are really used in Emacs Lisp.
2018 Update
It's been almost a decade since I wrote this post and the Lisp family of languages now appears to be gaining significant traction in the general programmer consciousness. Much of this appears to be related to Clojure which has not only become a properly separate dialect of Lisp in its own right, introducing many of its own good ideas, but also now has a near-identical version targeting JavaScript and has inspired many other Lisps targeting other platforms. For example, Hy targets the CPython AST and bytecode, aiming first for interoperability with Python, but using Clojure ideas "when in doubt." (Though from the latest commits, the latter may be changing a bit.)
The big change this brings in your decision-making process is that you should also be looking at whatever Lisps or Lisp-like languages are available for and interoperate with languages or platforms you already use, be it Perl, Ruby, Erlang, Go or even C++ on microcontrollers.
