Amt will point you in the right direction.
What I'm seeing all over this thread is a conflation of Greek military periods. Exomides would have been pretty common in the late 5th century (so, circa Peloponnesian War), but if you're portraying an earlier Spartan (like from the Persian invasions), you'll be better off in a chitoniskos. If you decide to develop that kit, a bronze cuirass would only be period-appropriate for a Persian wars era soldier. By the late 5th century, people started using tube-and-yoke spolas a lot more (though, Spartans were the last of the mainland Greeks to really hold onto bronze body armor, so as a Spartan, you may be able to fudge it).
It's a misunderstanding that Spartans wore red cloaks in battle. Indeed, Plutarch, the only one to talk about it (and several centuries after it happened), just says the Spartans "wore red in battles." Artwork and other written sources don't seem to indicate that Greek hoplites would wear chlamydes in formation (cavalry would wear them in battle, and I'm sure they were useful to all on the march and in camp). This is why most re-enactors who portray Spartans choose to have red chitoniskoi or exomides.
For the record, Plutarch also says that Spartans have a tendency to go around "without a chiton" all the time. Not sure if the implication there is that they would wear skirts, or if they would wear nothing.
The suggestions in this thread are all pretty good. The cool part about portraying a Greek is that it doesn't take a lot of time and money to be really, really accurate with it.