The most important things for getting into a science / engineering PhD program are your alma mater, GPA and GRE score. (I have no idea about arts programs, but since you mention research, I assume you're doing science or engineering.) In terms of alma mater, you don't need to have gone to MIT, but you should preferably be attending the flagship university in your state, or at least a good state university. If you're not, your GRE will be much more important. For GPA, you should be in the top 5 to 10% of your class to get a TA / RA at a top 10 school. If you're not "good" at standardized tests, practice the GRE until you can get a good score.
Research experience is critical if you are applying for fellowships like NSF, Covid or no. You'll be competing against kids from "open" states, so you're probably out of luck if you have no research experience. If you're a junior, though, you still have time. Find some research experience.
When applying, go and look at the research all the professors are doing at the university you want to attend, and contact two or three whose labs you think you would be a good fit for. If you meet GPA / GRE criteria, they'll circulate your application, and profs will have the opportunity to pick you up for an RA. This is very important and can make the difference between being funded or not.
Personally, my opinion is that if you are admitted without funding, you should skip the graduate degree and go to work. You can always apply again in a couple years. If you are admitted with funding, I see no major drawbacks in doing a graduate degree. You can always quit with a Master's if you realize it's not the path you want to follow. But you do need to be thinking about the career this is leading to. And "professor at top university" is probably not super realistic unless you are a superstar.