How to get nominated for awards? I came to the US with my PhD classmate having similar backgrounds and took postdocs at two different universities. My research output is much better than him, but he has won three awards from reputable societies/organizations. Now he got a tenure-track position, but I was not even invited for an interview.

The point is that his PI nominates him for every possible award, but my PI never does. I specifically asked several times, but he doesn't care.

What should I do? Should I move to another group whose PI supports me in getting such credits?
"My research output is much better than him". "Better" is not well defined in academia. No one has grades: you can't compare an A to an A-. Instead, everything is "graded" more casually. 

But this casual grade has nothing to do with how hard or difficult the work is. If you mean "better" as "I work harder, have more publications, and my publications use more difficult techniques etc." then I think you have the wrong definition of "better". In academia, "better" means your research is clear and interesting. Doing better in academia seems to usually be less about doing harder and more technical work, and more about picking the right problems to pursue and communicating the results well. This has nothing to do with how "good of a technical mathematician/scientist" you are, rather it has to do with how good you are at understanding what is interesting to a research community and clearly communicating results which reflects the interests of said community.

Your classmate is likely doing "better" because his PI is very interested in the results of his research. Others must be interested as well: that's why he got awards. You should look at his work and ask: who finds this interesting and why? See what you can learn about what his success says about the research community you are engaged in.

A good practice to do from time to time is to take a few papers from a high-impact journal and just try to understand why these are considered as highly impactful. What I have found is usually two things: 

1) They have a really clear introduction which motivates this work as part of some larger research question which many people are interested in.

2) They are simple.

The first is pretty simple: writing clearly is important. "Selling your work" is important because if people don't know at a glance that your work is interesting, they will never dig into the paper to find out. The second point is kind of counter-intuitive. Academia tends to be very elitist about "being smart", but a lot of the most impactful work is actually relatively simple in its methods and arguments. One reason this occurs is because a simple result is easy to generalize to other fields: biologists can follow clear and simple mathematical results, but a more advanced mathematical treatment is a subject for mathematicians, meaning breakthrough mathematical models without all of the extra bells and whistles are the results which are most widely known and applied. But also, people more easily understand it and see all of the nuances they can add to it, generating a lot research interest surrounding the idea (i.e. you left something for someone else to look into!). 

Of course, this depends on the audience. In mathematical journals sometimes very dirty mathematical derivations contain some of the most cherished results by other mathematicians. Knowing your target audience for each paper/presentation is the best way to make sure your research is read and appreciated.