Different processes work for different people and at different times.
Old Research Workflow Tools
OneNote - I used this in grad school with local notebooks, and briefly more recently with local notebooks synced through a third-party cloud service. It's not a bad piece of software, but I dislike its need to hook into Microsoft services (a constant annoyance with Windows).
Evernote - I used this for a year or so before the program's inability to competently handle a wide range of basic tasks and the seeming unwillingness to fix long-standing flaws (like tables and italics) in favor of merchandising and the pursuit of novelty drove me away. The lack of robust built-in privacy/security options was another persistent annoyance.
Nimbus Note - An interesting and very good Evernote alternative, but I never quite got comfortable with the UI. It looks like it's changed since then. Very friendly people here.
Notebooks - I think this is a single-person operation that saves notes as html files. It's fast and lightweight, and the developer is helpful and friendly. I still use this one for musical composition notes (not the actual notation).
.docx/.txt files - Simple, but more work to keep organized and less elegant for someone who doesn't typically keep a hundred browser or application tabs open at once.
Wunderlist - This was a great little app until it was bought by Microsoft. I exported my data and deleted my account the moment I saw the sale announcement. Of course, Microsoft then killed the product off anyway.
Current Workflow Tools
Standard Notes - Notes application with end-to-end encryption. It doesn't have the widest range of functionality, but what it does is executed extremely well. Friendly small developer team too!
Todoist - I used to use this and it's returned.
My Typical Process for a New Research Project
I first assess the state of the literature if I'm not already familiar with it. Key points are saved in a Standard Notes page and pdfs of important or useful papers are saved on SpiderOak.
I then write up something like an abstract to build a greater sense of direction for the project. This also helps me highlight specific issues that will need further attention. These are noted in Todoist.
If the potential project is still feasible, interesting, and potentially valuable, I make an entry in my research pipeline (.docx file) with two or three target venues.
At this point, I usually have a Word document with the abstract and a potential title. All files are organized into a project folder in SpiderOak.
Data-cleaning code is kept in one text file, while analytical code is kept in another. Data-intensive projects will often involve multiple files for different data-cleaning tasks and sets of models and figures. I tend to start things in Notepad++, but once I move beyond two or three code files, I create a project in Sublime Text and work from there.
Long-form notes are kept in Standard Notes until I have enough written that whatever notes are left are turned into comments in the document file itself. Any tasks are listed in Todoist so I don't have to remember them myself.