I'd say the best way to approach it is, firstly, to find the grape variety or blend that suits your palette. Secondly, try a few wines of that grape variety from different regions, be it New World or Old World, or just differing regions from within a single country.
My particular favourite is pinot noir, and i favour the robust, earthy reds of Burgundy - especially the Cote D'Or.
Now, pinot noir is grown all over the world, but the majority of wines from the New World made with this grape do nothing for me whatsoever! So just because it says pinot noir on the label doesn't mean i'm going to enjoy it.
Very generally speaking, New World reds (Australian, South African, etc) tend to be quite fruity compared to their European counterparts, which are usually a little earthier. But this is an gross generalisation and quite often you can find the opposite to be true!
I'm an Old World fan, so i'd say try maybe a good Barolo (Italy), a Burgundy pinot noir (France), or a good vintage Rioja (Spain). Look for DOC, AOC, VQDS or such on a label as this ensures that the area the wine was produced in is subject to quality controls. Most of all, have fun...and happy hunting!!!