Will I destroy my career if I published a paper with a serious mistake? I'm an undergrad and kind of new to this whole research thing. I've been doing research for the past ~9 months as a requirement to graduate with my bachelor's (there's the research track and software development track, I chose research).

I was told to try and submit my paper to conferences/journals (depends on their deadlines) to see if it gets accepted. But, one thing my professor said really stressed me out. If my paper gets accepted, and people find a serious mistake in it (ones that could cause your conclusion to be wrong, etc), it would destroy my whole career before it even began.

Can anyone with more experience go into detail about what could really happen? Assume, that the paper really gets accepted. On one hand, I'm not entirely confident of myself, and on the other, I have found a passion in research and would love to continue on for a PhD in future and this might help boost my resume a little given it is my only research experience.
Congrats on your paper. No, it wouldn't destroy your career but it would be awkward and embarrassing. It could potentially hurt your career if the mistake was the result of obvious sloppiness, gross incompetence, and worst of all, outright dishonesty. But for the first two of those, the damage would very likely be containable and if you keep doing research, after publishing another paper or two that had no mistakes, no one would remember this minor incident.

With that said, your advisor is correct that it's best to avoid publishing papers with mistakes in them if at all possible, so do make a sincere effort to check everything to the best of your abilities before submitting the paper.