I know a lot of graduate students go through this, so what are some tips to help get over this self doubt and fear of failure?

Start every day with a smile. Make use of your resources, which includes all of your labmates (grad students, postdocs, researchers, professors and advisors). These people are specialists and experts in whatever their little niches are, and they will be gone some day, so you better absorb whatever knowledge they have before they graduate or find a different job.

I feel a lot of self-doubt and I find myself avoiding the work for the group because I'm afraid of not understanding something and looking stupid.

In my first year, I was really nervous because I was taking a class that my advisor was teaching. I didn't do well on the midterm because I misread a question (effectively only answering half of the question.. the sad part is that I knew how to do it very well :/). After that complete fuck up, I felt so much better. I aced the final after that because I already knew I lost my chance to be completely perfect all the time in front of my advisor so I stopped feeling as nervous about making mistakes in front of him.

Recently, I corrected my advisor on some suggestions he gave me because I know my project slightly better (considering that he has to keep in mind several lines of research, it's expected, I guess). But correcting him and him acknowledging that correction have made me feel even less nervous about what I'm working on.

So, basically, my suggestions are:

(1) use your resources to learn enough to become awesome

(2) fuck up in front of your advisor early because you'll feel less pressure from trying to be perfect and your fear of failure will be weakened

(3) correct your advisor because then you'll realize that advisors are humans who also have failures (but, you know, they seem smaller because they've learned to not make as many mistakes or, at least, less big ones)