Does someone know in which books, lecture notes or ... I can find the induction step of the proof of Hilbert Syzygy Theorem? I'd only found the proof for R[x] (e.g. Weibel) and I haven't really an idea how the induction step works. Thank you!

I'm not sure exactly which kind of proof you're looking for, but a proof by induction using Gröbner bases is presented in Chapter 15 of Eisenbud's Commutative Algebra, see Corollary 15.11 specifically. There's also a more abstract homological proof in Chapter 19, see this related question: Best exposition of the Proof of the Hilbert Syzygy Theorem by EilenbergCartan The first proof is closer in spirit with Hilbert's original proof. 


A proof using Gröbner bases is in Using algebraic geometry by David A. Cox, John B. Little, Donal O'Shea, Theorem 2.1. However, I was always sure that there should be (at least in the graded case) an inductive proof along the lines of AtiyahMacdonald's proof of HilbertSerre theorem, namely by induction considering the 4term exact sequence $$0\to K_i\to M_i\to M_{i+1}\to L_i\to0$$ where $K_n$ and $L_n$ are the kernel and the cokernel for the operator of multiplication by $x_n$ (these are modules over the polynomial ring in $n1$ variables), but something escapes me at the moment, so I just leave it here as a wish.... 

