Learn Russian: Это большой проспект.

When you use Это to mean "This is" or "That is", it acts as a fixed starting word and stays exactly like this, no matter the gender of the noun that comes after it. You only need to match the gender if you are saying "this avenue" (этот проспект).

Where is the word for "is" in this sentence?

In Russian, the present tense of the verb "to be" is almost always left out. So, the sentence literally reads "This big avenue", but it means "This is a big avenue".

How can I tell that проспект is a masculine noun?

You can usually tell the gender of a Russian noun by its last letter. Because проспект ends in a consonant (the letter т), it is a masculine word.

I learned that masculine adjectives usually end in -ый. Why does большой end in -ой?

You are right that many masculine adjectives end in -ый (like новый for "new"). However, when the spoken stress (the vocal emphasis) falls on the very last syllable of the adjective, the ending becomes -ой.