What is the subject of this sentence?

Contrary to what others have said, the grammatical subject is "There".

Sentences of this form are called existential clauses, and "there" is the subject (it's called a dummy subject).

See page 249, A Student's Introduction to English Grammar http://books.google.co.uk/books?id=qlxDqB4ldx4C&pg...

Edit: <sigh> read the goddam book reference before marking down. The above book is written by the main authors of The Cambridge Grammar of the English Language ( http://www.cambridge.org/uk/linguistics/cgel/ ). Who's more credible? Them, or some unsourced opinion in an Internet forum?

But doesn't the subject of a sentence indicate what the sentence is about? That being 'things' in this case? Don't have an English major but just wondering.

Things

The subject is "things". Your verb is "are", and there "are" things. Plenty describes the number of things.

"Plenty" is your subject. "Things" is not your subject because the subject of your sentence will never be within the prepositional phrase ("of things," in this case).

"things" because your talking about those objects making them your subject

the subject is 'there'; it can stand alone w/it's verb & adverb