Fields are one of the following: scalars, vectors, spinors or some Lie algebra elements, right? And it's often said that scalars are spin0 and vectors are spin1. So, what's idea of correspondence between nature of field and it's spin value?

A classical field is a section of a vector bundle on the spacetime manifold $M$. That vector bundle is typically obtained by using the associated bundle construction applied to the frame bundle of $M$, and some irreducible representation of SO(d) (you've noted that I'm in Eucledian signature). Sometimes, this is not enough, and one has to start with a double cover of the frame bundle of $M$, and use some irreducible representation of the double cover Spin(d) of SO(d). So, the types of field (scalar, vector, ...) are in onetoone correspondence with types of irreducible representations of the spin group Spin(d). Those irreducible representations are classified. spin0 = 1dimensional trivial irrep For d=4, we have an isomorphism between Spin(4) and SU(2)xSU(2), and so irreducible representations are classified by pairs of nonnegative halfintegers. 

