Should one learn point-set topology before real analysis or before studying metric spaces a bit? There are some advantages to doing so -- a more unified approach to real analysis or the study of metric spaces, for example. But this comes at the cost of all motivation for point set topology.

One can do "classical" algebraic geometry rigorously, and this is not a bad idea. It provides much-needed motivation for the language of schemes, sheaves, etc., which can otherwise seem incredibly unmotivated. And it generates intuition (which, to be fair, is often wrong) about these complicated and often pathological objects. But an even better reason to study classical algebraic geometry is to discover why Grothendieck, Serre, etc. wanted to come up with modern algebraic geometry in the first place; it's because classical algebraic geometry is so obviously in need of fixing. It's a beautiful subject, but I think it's pretty obvious from even a short study of it that you're not getting the whole story. (Bezout's theorem is a great example.) A good book to read if you want to get this feeling is Harris's "Algebraic Geometry, A First Course" -- it's a well-written book filled with great motivation, but you can't help but think that it's holding something back.

As for commutative algebra -- I think it makes sense to learn it concurrently with the geometry, which motivates it in an incredibly compelling way. Eisenbud is a good place to go for that kind of motivation.

I'm sure others will address the issue of doing research in the subject; I'm not qualified to comment on that, beyond mentioning that there's a wide variety of subjects researched in the field. Many people still work on subjects that might be considered "classical."