I'm an undergraduate biology and chemical engineering dual major student. I recently got into pure math while I was researching patterns in biology which lead me down the rabbit hole of fractals and non-Euclidean geometry and eventually, lead me to number theory. I really enjoy the subjects. Is it possible for me to pursue my graduate studies in pure math? If so, what would be the necessary steps I should take to do the same? I am in my second year of studies and I do intend to take the GRE Math Subject test but I would really appreciate any other advice you could give me. Also, switching majors is not an option because my uni doesn't have a good math program. If you are reading this, I hope you have a nice day ahead.
Regarding graduate programs in the US: It is possible to get into graduate school for pure math with a non-math undergraduate, but it might not be the best idea if you have the option to change your undergraduate course of study.
In the US, graduate programs in pure math will typically expect you to have taken coursework in differential equations, linear algebra, real analysis, and abstract algebra. They may also expect you to have taken at least some coursework in other upper-division (third and fourth year) topics. Once you've taken all this coursework, you're well on your way to graduating with an undergraduate degree in math!
When reviewing applications, a program wants to make sure that their incoming graduate students have the potential to complete their studies in a timely manner and produce good work. (Graduating in a timely manner ensures that the program's budget stays balanced, and producing good work can help you get a job that reflects positively on your graduate program.)
Your coursework and GPA (and maybe your GRE scores to a lesser extent) reflect on your potential to graduate in a timely manner and produce good work. However, as many answers on this site will tell you, letters of recommendation are one of the most critical ways for graduate programs to get a sense of your preparation and potential. The people who are best suited to discuss your potential as a future mathematician are other mathematicians, so you want your letters of recommendation written by mathematicians if at all possible. This means you should be meeting with math professors and discussing math with them, which will be much easier to do if you are focusing your undergraduate studies on math.
I didn't become a math major until my third year of undergrad, although I did take a few math courses beyond calculus before then. However, the math professors that I spent time interacting with as a math major made all the difference in helping me get into graduate school. They gave me advice, worked with me on reading projects, and encouraged me to apply to schools that were a good fit for my interests and abilities.
I wouldn't worry too much about your university not having a good math program -- a top-ranked biology or engineering degree isn't going to be as helpful as expert advice and support from mathematicians, even if they work in a department that doesn't "rank highly." Even if you can't change your official course of study, I would encourage you to take more math classes before setting yourself on graduate school in math. You might find out that you don't like math enough to spend 5 years on it, or you may find out that you like applied math more than pure math.