I am also an undergraduate hoping to start a career in quantum computing someday. I'm a physics student who became interested in the subject about a year ago, and these are some things that helped me build a foundation.
In terms of background, linear algebra is the only course that is essential for understanding the basics of the subject. The reason is that computation can be simplified as a series of matrices (called gates) acting on a particular vector (called the state). A course in quantum mechanics will be necessary for more advanced studies and many applications of quantum computing, but you are perfectly qualified to start learning the basics without such a course.
In terms of resources, my advice is to start with something lighter than a textbook for your first introduction. I highly recommend the Microsoft Q# Support Docs, especially the "Quantum Computing Concepts" articles found here. If you're looking to start programming, Q# will be very difficult to learn without background in C# and a functional programming language, so it may not be the right language to start out on. It's nothing against the language, but it was hard for me since I had never used C# and had trouble reading the language-specific docs before my functional programming course. I personally have a lot of training in Python, so languages like Google's Cirq or IBM's Qiskit were more natural choices for me.
Once you've gone through a few of those articles on the basics, that's when I would pick up a textbook. Someone has already mentioned "Mike and Ike" (Quantum Computation and Quantum Information by Michael Nielsen and Isaac Chuang) which is one of the most highly regarded books on the subject. Another I'd like to mention is Quantum Computer Science: An Introduction by N. David Mermin, which is pretty accessible for someone without a background in quantum mechanics, at least for a few chapters. No book is going to be right for everyone, so just try a few and see what makes sense for you.
My last piece of advice is to find a friend to work through material with or a professor to help walk you through particularly difficult topics. Something to remember throughout your academic career is that math, and related fields, is better with a guide.