I stand by my answer 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.