Just citing what they cite I’m not a professional academician by any means. I’m just a medical student trying to write a paper in my spare time.

I don’t have an excellent grasp of the literature in this field. A lot of the information I’m learning comes from a single excellent review article/book chapter.

Here lies my question. This book chapter will say, e.g., ”such and such [1][2]”. Do I just cite the book chapter when I describe such and such in my background? I think not; I should probably cite [1], [2], or both.

So, the real question: is it OK/acceptable/ethical to just believe that [1] and [2] contain such and such; and just cite them without actually reading the papers? To be able to do this would save me quite a bit of time.
You need to make an effort to get and understand the work related to yours. That is the only way you can make sure you are building on them, not misunderstanding. If you cite it, it means to me (as a reader) that you did use that work to reach your conclusions. You might get away with citing indirectly stuff you didn't check in detail, i.e., "Smith [19], summarizing previous work by Adam [1], Eve [2] and Noah [3, 6], states that...".