'In high school I was taught that the bourgeoisie were society's middle class.'

What you've got here is a problem of definitions. I'll clear this up first before addressing your main question.

Bourgeoisie: Marxist theory defines the bourgeoisie as those who own private property. In Marxist terminology, private property is productive property, like factories, mines, oil-wells, farms and so on,that is owned by a single individual or a small group of individuals, as opposed to being owned by society as a whole. By owning such property, the bourgeoisie are able to acquire wealth (in the form of profits), which they did not create and are not entitled to, by paying their employees less than the value they create (this is called Surplus Value).

Proletariat: The proletariat is the opposite of the bourgeoisie in that they are people who do not own private property. They therefore do not have a permanent objective source of income (profits from a business) to fund their lives, but must instead sell their labour to the bourgeoisie. In other words, in exchange for their work they are given a wage. A proletarian, therefore, is someone who sells their labour to another.

Middle Class: The general social definition of 'middle class' is far less precise. But generally it is considered to mean people who: are reasonably well educated, such as having a degree from a good university; work in a technical or professional job, perhaps as a doctor, lawyer or middle-ranking civil servant; and enjoy a comfortable standard of living, i.e. they live in a nice neighborhood (perhaps the suburbs), don't have to worry about paying the bills, and can afford luxury goods (big TV, nice car, foreign holidays).

The definitions of bourgeois and proletarian can overlap with middle class. Allow me to demonstrate this with two examples:

First, imagine a man called John. John owns a small factory that makes widgets. His business is reasonably successful, it is profitable enough to provide him with a comfortable middle class lifestyle. He has a fairly big house, he drives a nice car and goes on holiday to Spain once a year. John makes his money by extracting surplus value from his employees. John, therefore, is middle class and bourgeois.

Now, imagine a woman called Joanne. Joanne is a doctor, with a medical degree from a well respected university. She lives next to John in a similarly nice house. Her car isn't as nice as John's, not because she can't afford a nice car but because she isn't interested in cars. Instead, she loves art and spends some of her disposable income on collecting it. She also goes on holiday, but she prefers visiting France. Joanne works at a private hospital; she sells her labour to the company that owns the hospital (and the hospital profits from this). Joanne is thus middle class and proletarian.

Clearly then, the proletariat and the middle class are not enemies because they are not mutually exclusive groups.

I hope this answers your question, although I'm sure it also raises more questions. Please feel free to come over to r/socialism to ask those follow up questions, as that's a more appropriate forum for political discussion than here. :)