I always prefer to have some exposure to material before I take a math class. For me, part of learning math is putting the ideas in my brain then letting them simmer for a while. The longer I give myself to do this, the more densely interconnected the ideas have the opportunity to become.
Plus, if you've already grappled with the surface questions of a subject then you can spend the semester mulling over the subtleties instead.
I can't comment on what pace is "best," but I'll say this: I go on occasional benders of trying to push myself forward as far as I can manage, if only to identify what concepts I need to strengthen before I can approach newer ideas. The deeper I go in math, the more enjoyment I get from connecting ideas that I originally didn't realize were connected...but in math, it turns out that lots and lots of the ideas are.
For me, motivation comes from having unanswered questions, but everyone will have different opinions on what kind of unanswered questions are even interesting. At your level, though, a book on discrete mathematics might make for enlightening reading. It's the kind of math that you can count, you can write down...meaning it's not terribly abstract, so you can really grapple with the ideas without them feeling vague.
A few years back, I spent the summer reading the chapters of a discrete math book (Gossett, Discrete Mathematics with Proof) that hadn't been covered in my discrete math class the previous semester, then I studied through a another discrete math text (Kolman, Busby, Ross, Discrete Mathematical Structures) that had some chapters with introduction to groups (which are something discussed in abstract algebra), and that gave me the confidence to start reading an abstract algebra book (Pinter, A Book of Abstract Algebra) that I had picked up. For me, the ideas of abstract algebra give me a lot of motivation, because the concepts of homomorphism and isomorphism are the sort of thing that start to show up all over the place, when you're looking for them...and that's the sort of thing that makes me enjoy math: Recognizing ideas in places where I didn't know the idea would be lurking.
Once I had those concepts of homomorphism and isomorphism in my head, I studied through my linear algebra book (Anton, Rorres, Elemnetary Linear Algebra) again, this time feeling like I was getting a much more meaningful story. And this left me feeling prepared to read through a book on applied analysis (Holland, Applied Analysis by the Hilbert Space Method), which is the sort of math that really keeps me engaged with the subject. (I'm looking to do mathematical physics in grad school and beyond...so if your interests with math are up a different avenue, your path may be substantially different.)
Neither discrete math nor introductory linear algebra require calculus per se, although calculus ideas tend to pop up in the textbooks (but often as optional sections). What might make linear algebra difficult at your level may be the tendency to only think of the geometric interpretation of linear algebra. (It is, after all, the math used to do 3D video game engines and whatnot.) But it's the abstract side of linear algebra that makes it such a compelling subject, like what happens when you say to yourself, "I need to stop thinking about vectors as 'pointing in a direction' and 'having a length' and all that, and ask why it is that the math works when I take a group of functions and call them vectors, then interpret the concepts like orthogonality and projections." It was these sort of questions that, for me, really started to raise deeper questions and provoke a deeper interest in the subject.
