These are my personal opinions. Other people will say other things.
You must apply for fellowships to be competitive. It actually doesn’t really matter if you get one because you find out after you are admitted to the schools. But you need to be able to say on your application that you applied (they specifically ask). If the school thinks you might win one then they will be more likely to admit you because they won’t have to pay for you. The NSF deadline is /earlier than you think/. It is usually late October or early November… which means you need to ask for letters of recommendation in late September or early October.
You should at least apply to:
and probably to:
When I was doing this as an undergraduate, it took me 4 to 6 hours per paper to read it, understand it, and come up with a question.
One thing that I found worthwhile was to read some of the sites out there on getting into grad school. I liked: