Observation I. It’s damn-near impossible to find someone who gives a straightforward definition of an algebraic variety straight off the bat.
Instead, most authors tend to define an affine algebraic variety – first as the common zero set of a collection of complex polynomials in and later as a “variety that can be embedded in affine space as a Zariski-closed set” (Smith et. al., An Invitation to Algebraic Variety). Then, half a book later or more (it’s on page 144 of the aforementioned book), it’s said that an (abstract) algebraic variety is a topological space with an open cover consisting of sets homeomorphic to affine algebraic varieties which are glued together by so-called transition functions that are morphisms in the category of affine algebraic varieties.
This of course requires knowledge of category theory, the Zariski topology, etc. etc.
As of now, this 30+ minutes of searching has gotten me through about 3/4 of a page in Chris Elliott’s online manuscript concerning -modules.