How to be confident that my background for my research is adequate?
To expand a bit on the comments saying this is a normal feeling (which it is): learning something in a class is well and good, but many things will not truly click until you have to use them. To take a random example, I learned and understood the proof of the Arzela-Ascoli Theorem in my third year of undergrad, but I thought it was just a neat result. I certainly forgot the statement and proof multiple times, and had to look it up. It wasn't until halfway through grad school that I realized this theorem (and/or the idea behind it) underlies pretty much every compactness theorem for function spaces, making it (in my judgment) one of the most important theorems in analysis. After that, it was not so hard to remember.

Due to this to me it feels I should read a textbook for 2 weeks then start doing the research.
Others have suggested to only read a book when you need a particular lemma/proof/etc., however I am conflicted on whether this is good advice.

Some subjects you'll need to know in detail, others you can pick up on the fly. With experience, you'll get better at distinguishing one from the other, but for now your advisor can give you guidance about this.