This is a personal recommendation. There are probably many things that you could do/read/study that would help you in your journey into the world of pure mathematics, but I have one suggestion.
I suggest getting a good book on abstract algebra. A beginning book on abstract algebra will usually cover the topic of groups. A group is a set with a (binary) operation such that certain axioms are satisfied. One hard thing about abstract algebra is that it is ... abstract/pure. But it is also simple (no pun intended). You don't need a lot of background to get started. The background required is some familiarity with mathematical logic and set theory. It also helps a lot of you have studied linear algebra. How do you find a good book? You could go to the library and look at their section on abstract algebra/group theory. If you do not have access to a university library, you can often get your local public library to get some books for your from other libraries. I would get a stack and look through the first sections in each book. You can, of course, also search for free notes online.
Basically I would suggest that you try to undestand what a group is. Look at the definition and simple go through a bunch of examples. Maybe you can make a little notebook with one example per page. For each example check that the group meets the definition of a group. Then also find examples of sets with operations that are not groups and make a list of these.
After this you would try to understand how groups of smaller order have to "look like". As you learn more about groups you could determine whether the groups in your notebook are Abelian or non-Abelian, cyclic, finite or infinite.
Anyway, the point of all this is not just to teach you about groups, but to get you used to thinking abstractly. Specifically, how can I check (abstractly) that something meets a definition.
