Are operators also considered to be keys? I'm taking a MongoDB course and one of the assignments asks to include quotes around keys. 

So I did for a query like this {$push: {"key":value}}

But I was told that $push is also a key because it's on the left hand side. I always thought keys were like identifiers for values whereas operators are just actions, they don't have any value associated with.
"$push" is a key in the JSON context. Don't forget everything in mongo (queries, projections, documents, method parameters) are represented as JSON objects. So, although $push is an operator for Mongo, it is still a key on the JSON object it belongs to.

JSON's specification demands key names to be surrounded by double quotes, although the Mongo shell, which is actually a Javascript interpreter, is more flexible and allows key names (field names and operators) without them, as the language does.