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Hello, this question may seem specific, what I'm asking is are there any resources to learn more about what a law firm expects you to know about:
1: Client Confidentiality
2: Bank and Securities

I'm looking for books/readings to gain a better understanding of these topics.

Client confidentiality is one of the central pieces of professional responsibility; all law students learn it in law school classes (and some clinics), it's required for the Multistate Professional Responsibility Exam, and also tested on some state bar exams. Check out materials for professional responsibility courses, or the model rules of professional responsibility, for comments on that. All law firms expect you to be familiar with this concept and adhere strictly to it.

Banking and securities law is much more specific; some law firms practice in those areas, and others don't. So there's no general "guide for what a law firm expects you to know about" these topics.

Alright, that's sort of what I expected. I was hoping for something along the lines of a NOLO guide or something for some basic understanding/guidelines. Thank you!

What perspective are you asking this from?

If you haven't taken a professional responsibility course in law school before you start working/interning, you should say something to the attorneys and ask if there are any resources you should review or any overview they can offer. Generally the "difficult" things in professional responsibility will not come up for someone who is just interning or something, basically just keep stuff confidential.

As for "banking and securities" - these are hopelessly broad topics and make me think you are not in law school yet. This answer is not one that can be given. You will not be expected to really know anything and you will be expected to learn things on the job.