I prefer to start as soon as I know what I need to research. With my current novel I made the mistake of waiting to research underwater archaeology until after I started working out my story. Not smart. My research not only made plot changes necessary but enticing. Based on your characters, setting, and plot, you should know from the beginning some of what you need to look up. I make a research list toward the beginning of the project and check off items as I go. It includes major and minor research points so I don't forget.
Organizing all that research can be difficult. Part of the point of research is to have something to refer to while you're writing. That means it needs to be easy to access. What do you use for research primarily? I use the Internet and library or bookstore. But research can also encompass visiting locations and interviewing experts, depending on your topics. For Internet research, I label folders in my favorites with my research topics. I save any and all websites that will help me (you may never find them again otherwise) into those folders. You may want to nest all research topics under your book title. Some topics I know I will use across different projects so I don't do that. For easy reference, I also take "notes" by copying and pasting information from the website into a file that's saved as a specific topic. I make different documents for different subjects and save those in a folder nested within my book's general folder.
Books are a different matter. I highlight as I read if I own it and/or take notes and keep them together either in the same notebook or file folder. I make sure I note where the information came from so I can find it again. I use photos for research a lot, whether my own or photos I've found online. I save them in a specific folder and label it according to the subject.
These methods have worked great for me, but you have to find your own rhythm with research and organization. Work with your preferences and tendencies, not against them. Experiment with different ways of handling research and see what works best. Once you find your groove, you'll be in a great place to get writing.
Photo by pathlost