Is Constructivism (philosophy of mathematics) against classical logic? I might be wrong, but mathematics' main branch of logic is based on classical logic, and I was wondering if Constructivism was against that main branch of logic, or it wasn't against that main branch because its views were not directly opposed to it in any shape or form. Also, if Constructivism allows us to get rid of the Liar's paradox, doesn't that mean that classical logic is flawed and there's a more flawless logic than classical logic?
Mathematical constructivism typically involves the use of 'intuitionistic logic', which avoids LEM (Law of Excluded Middle) and which is central to classical logic, so in that specific sense, the two are incompatible. Not relying on LEM means that constructive mathematics can be regarded as one sort of 'generalisation' of classical mathematics, in the same way that not relying on Euclid's fifth postulate results in a generalisation of Euclidean geometry.
That said, one can do constructivist mathematics without being inherently opposed to LEM - it might be, for example, that one is seeking a constructive proof for computational reasons, rather than for any particular philosophical reasons.
i'm not sure what you mean by your reference to the Liar's paradox, as there have been a few different resolutions proposed: cf. the Stanford Encyclopedia of Philosophy entry on it: https://plato.stanford.edu/entries/liar-paradox/#SomeFamiSolu
Personally, i think it's not useful to conceptualise things in terms of 'more flawless' logics, as different logics can have different uses (cf. e.g. Wikipedia's list of applications of paraconsistent logics, https://en.wikipedia.org/wiki/Paraconsistent_logic#Applications), and are interesting in their own right regardless. Andrej Bauer has a post about the many worlds of mathematics: http://math.andrej.com/2012/10/03/am-i-a-constructive-mathematician/