First thing to note is that 300 did the shields wrong on purpose. Zack Snyder figured that all-metal shields with the double grip across the middle would look cooler than the real thing. I can't really fault him for that, as he was making a movie, not writing history. But don't be distracted by what you see. (I've written more about the inaccuracies of 300 here.)

The actual hoplite shield (called aspis) was made of alternating layers of strips of wood. It had a double grip of which the loop for the elbow was in the exact middle of the shield, with the loop for the hand all the way on the right edge. The rim of the shield was reinforced with bronze. Some shields were entirely coated in a thin (<1mm) layer of bronze, while others simply had a wood facing. Either could be decorated with paint. Before the Persian Wars, every hoplite could pick whatever blazon he wanted for his shield, but afterwards some cities appear to have gone for a standard symbol or letter.

Spartan shields were known for two things: one, they were all covered in bronze, and two, they had the capital Λ on them, which stood for Lakedaimon (the region Sparta was in). Xenophon and Plutarch both claim the Spartans polished their shields before battle, and Xenophon likes to stress how the Spartan phalanx looked like a single mass of red and bronze, which means the shields must have had a bronze coating. Other sources report that they could be recognised from their lambdas. Given the references to shining bronze, it is clear that significant parts of the shield would have been unpainted. To my knowledge, no source says anything about the colour of the lambdas - I'd be glad to be proven wrong on this if you can find the reference. It is possible that the lambdas were red; it is also possible that they were another colour; it may even be that the shields were red and the rims and lambdas unpainted bronze.

The interesting thing is that the earliest references to the Spartans' famous lambda shields (in Eupolis) and to their red tunics (in Aristophanes) date to the late fifth century BC. In his detailed description of the battle of Thermopylai, Herodotos says nothing about the nature of Spartan shields or clothing. Most likely the Spartans simply did not yet have their iconic outfit at the time of the battle (480 BC). Each Spartan will have been dressed according to his own preference and the blazon on his shield will have been an emblem of his choice. It may be accidentally true to history that the modern statue of Leonidas set up at Thermopylai (made in 1955) doesn't feature a lambda on the shield, but a gorgon's head.

Since the Spartans did not have uniform equipment before the Peloponnesian War, it is impossible to identify them on vases. I know of no vases that unambiguously portray Spartans, unless there is direct reference to Menelaos in Homeric combat scenes. There is only the extremely questionable theory that the warriors on the infamous Chigi Vase are Spartans because they are accompanied by a piper, and Thucydides says the Spartans marched to the sound of pipes - two and a half centuries later. I personally do not buy this at all. There really is nothing we can say about Spartan shields from pictorial evidence unless we go in for a circular argument ("Spartan shields are red; the shields on this vase are red; therefore these are Spartans").

tl;dr: we know much less about this than we would like. The probable answer is that Spartan shields were the colour of their bronze facing, with lambdas painted on them in an unknown colour.