I have some idea of what the main concepts of calculus are about but I have never actually taken a calculus class or studied myself. My general understanding is that before the concept of limits were formally defined by later mathematicians, the first people that actually invented calculus explained the idea of derivatives and integrals using the hyperreal numbers, something for which they were actually criticised by some for the concept being more intuitive rather than mathematically provable/valid. I also believe that modern, standard calculus does not deal with hyperreals at all but rely solely on real numbers in explaining all of its concepts.
So I was thinking whether it would be a good idea to get the gist of what hyperreals are and how they relate to limits as a foundation before actually taking on calculus, or would it be not worth it. I'd appreciate any comment/suggestion you guys would make.
"the first people that actually invented calculus explained the idea of derivatives and integrals using the hyperreal numbers": This is not strictly speaking true. Leibniz, for example, did not use "hyperreal numbers". He did use infinitesimal numbers, and made it clear in a 1695 publication (a response to Nieuwentijt) that such numbers were in violation of the Archimedean property.
Whether or not "it would be a good idea to get the gist of what hyperreals are and how they relate to limits as a foundation before actually taking on calculus" really depends on your goals. One advantage of learning the infinitesimal approach first (using for example Keisler's textbook) is that it will give you a direct intuitive access to the key notions of the calculus such as derivatives, continuity, and integral. Once you understand these key notions, it will be easier to understand the epsilon-delta definitions thereof that are usually presented in the (non-infinitesimal) calculus textbooks. However, if your goal is eventually to enroll in a university calculus course, you should be careful about actually using infinitesimals on a test (or even in the classroom): this may be misunderstood and will not necessarily facilitate your academic progress toward a degree.