I stand by [my answer](http://mathoverflow.net/questions/31650/modern-algebraic-geometry-vs-classical-algebraic-geometry/31730#31730) to the question that you linked.   In particular, I think that the distinction between "classical" and "modern" algebraic geometry is a little artificial, and I don't think that anyone is *meant* to do any particular thing; 
what you need to know depends on what theorems you want to read/use/prove.

But whatever direction you ultimately intend to pursue, it makes good sense to learn
varieties first.  As well as Chapter I of Mumford, there is Chapter I of Hartshorne, and
its many exercises.  The first few sections in particular are crucial.  There is also
Griffiths and Harris, which has no mention of schemes, as far as I can recall, but an awful lot of algebraic geometry of varieties.