Newform theory for $\mathrm{GL}_n$ was originally developed over non-archimedean local fields (at least for $n\geq 3$). The local statements readily yield their global adelic counterparts, since a cuspidal automorphic representation is uniquely a restricted tensor product of local admissible generic representations. It is then straightforward to translate from adelic language to classical language.
So all you need is newform theory for admissible generic representations of $\mathrm{GL}_n$ over a non-archimedean local field. The guiding questions are: what is a newvector in this context, to what extent is it unique, how does it generate the representation, and so on. This theory was developed by Jacquet, Piatetski-Shapiro, and Shalika in their paper "Conducteur des représentations du groupe linéaire", Math. Ann. 256 (1981), 199-214. The paper also has a corrigendum.
A more accessible reference is Section 13.8 in the textbook of Goldfeld and Hundley "Automorphic representations and L-functions for the general linear group". In general, reading this book should clarify many of the questions you might have. It filled an important gap in the literature.