Doubtful. I don't think any classical Greek (i.e. athen, mace, spart) cavalry had shields. (Greek pottery tends to depict mounted hoplites, who fought on foot.) The situation is different for non-Greek cavalry (Paeonians serving under Philip and Alexander where known to have shields).
During Hellenistic times (i.e. ptol, rome, sele) some Greek cavalry types adopted (large) shields (aspis or thureos, not peltē). Horse armour appeared as well, though probably the exception rather than the rule (hence reserve for ptol and sele champions).
As for classical Greek horseman equipment, the most important source is probably Xenophon, On Horsemanship 8: http://perseus.uchicago.edu/perseus-cgi/citequery3.pl?dbname=GreekTexts&query=Xen.%20Eq.%2012&getid=1
Here are also four relevant pages from The Cambridge History of Greek and Roman Warfare (Cambridge 2008):
And an ancient fresco of a charging hetairos (Macedonian companion cavalry):
How about the following?
basic: tunic, hat
advanced: simple helmet, long sleeves, perhaps some body armour
elite: helmet, boots, body armour, pteruges, perhaps left arm protected by a cheir (arm-guard, Latin manica)