Her teacher is also interested in your response, as she has two sets of materials, indicating two different definitions of what counts as an adverb. As Mary Poppins might have said, adverbs are as adverbs do. What matters is their function in the sentence, not their form. Adverbs modify not only verbs, but also adjectives and other adverbs. Furthermore, an adverb need not be a single word.
Some prepositional phrases are adverbial.
This is because they answer adverb questions. Please note, however, that today does not have to be an adverb. In this sentence, today is the subject of the sentence. Let's play with your sentence a little to show you what is going on. Today, I am going to play soccer.