How would I make this? I'd like to make a clear, shiny plastic material. Is this possible?

Yes, you can

My suggestion would be:

Treat the material as if it were glass (ish). Use the reflection/refraction sockets to get what you want.
Sorry, I can't be more specific that that at present 'cos a) I'm using a different PC than my home comp, and b) my memory sucks.

But it can be done:-))

Cheers,
Diolma

(Maybe someone more knowledgable will chime in..)