Mistakes?
(1) By accident, I had a relatively good career going around DC in applied math and computing for US national security. I thought it would help my career to know more math so got a high quality Ph.D. in applied math. I WAS then better qualified, but I also had broken my career continuity. My brother saw that early on: "After your four years in graduate school, you will be four years behind in your career."
And, I didn't realize that the opportunities in DC were unique or nearly so. I would, could, should have (A) not assumed my career would work away from DC and (B) stayed in DC.
Lesson: If you have a good career going in DC, NYC, Silicon Valley, wherever, don't assume without good evidence that that career will also work in another location.
(2) Near the end of my Ph.D. work, I had a part-time job and did some good things. Then occasionally a person never seen before was in the office for a few hours one or two days a week and wanted to chat with me about whatever, even sports. Eventually he got my opinion on what the US did in Viet Nam. In retrospect I was being recruited for some high end highly secure slot but violated a rule Mom gave me -- never talk about politics, sex, or religion. I had talked about politics. Don't do that. So, being frank and honest about Viet Nam cost me a good career step.
(3) Dad's idea about a career was to get a good education and, then, a good job based on that education. Eventually I concluded that, nope, instead, with or without a good education, it is from important to crucial to own a business and to make it successful, and early in a career near the top of the TODO list is to find a way to own and run a successful business.
(4) I didn't understand an important point about career dynamics: (A) It is tempting to guess that academics is good preparation, maybe the best, for a non-academic career. (B) Instead, for a successful non-academic career, it's important to ride the waves of the hot topics, and, bluntly, academics is largely out of touch with what the hot topics are. (C) While some academics can help, possibly a lot, there is a special way to know what the hot topics are in a non-academic career -- see what work and topics people are getting hired for.
Net, be reluctant to let academics tell you what will be important in a non-academic career and, instead, let a non-academic career tell you what topics are important for you in academics. And if have a good ugrad education, then pursue those topics while on the job by independent study evenings and weekends, maybe partly on the job -- to pursue those topics, don't leave the job to become a full time student.
E.g., when I was in my Ph.D. program, there was a prof, soon famous, who apparently had been hired as a consultant to work on a topic, call it, X. Well topic X was in practical computing, a bit tricky, not at all a topic in academics, and from my non-academic work a topic I knew well. With resentment the prof asked me: "I suppose you know X?" I had a one word answer: "Sure". Really it was a collection of topics such as X that had my career going well, e.g., annual salary six times the cost of a new high end Camaro. For that summer job working on topic X, the prof was about to have a tough time!
(5) One possibly good career direction is to get a job in a startup in its early days and get stock. For the stock, before joining (A) get the stock deal in WRITING, (B) study the details of the stock deal (e.g., vesting), and (C) have the deal reviewed by an appropriate lawyer. Handshake deal? Don't do that.
(6) Applied math and, usually, computing work or don't independent of human emotions. People, however, commonly are driven more by their emotions than what is true in math or computing. It is crucial, then, to pay close attention to the emotions of people.
As a special case, one way to get people to hate you and work against you is to present some work in math or computing that (A) has power important for the organization, (B) is understood only by you, and (C) looks rock solidly correct and, thus, not easily ignored. Then the people who hate you maybe cannot find fault with your work but might be able to arrange to get you fired.
Part of one way to get you fired is gossip -- be aware that you may have to defend yourself proactively, guess what you are secretly accused of and then defend yourself against the accusations.
More generally, whenever you start to be especially successful, you will attract attacks like some piles attract flies.
(7) There is discrimination in the US. Not often mentioned is that there are some cases of severe discrimination in the US against native born Christian males of European descent. So, it can be important to realize that fact.
(8) Try to keep confidential and not to report on work in progress: If your work looks good, your enemies will try to kill off your work before it starts to show its power or arrange to take credit.
(9) There can be corruption, e.g., people getting quite secret financial kickbacks. So, you can get a bitter enemy if, even without realizing it, you threaten their corrupt money. So, at least try to guess who might be corrupt and how, and then try to avoid that person or their source of corruption.
Of course, if there is someone corrupt in the part of the organization chart you manage, you will likely want to get this person moved out. Of course if the corrupt person has some close, say, family, connection with, say, a member of your BoD, then you have to be careful!
Point: Be aware that corruption exists.