HRTEM, HAADF and EDX is the best option for my personal point of view.

We've used depth-profiling XPS for this, which isn't straightforward! It requires a variable photon energy at a synchrotron facility. Yes start with EDX or XRF to check you see you have sulfide and selenide. Also you can look at UV-VIS and PL to check that lifetimes are being extended as intended. And yes TEM before and after to see the size changes. But to know if you have distinct core/shell structure of sulfide and selenide, without alloying etc., you need XPS I think. Changing the photon energy changes the smapling depth, so you sample more or less of the core and shell and see how S and Se signals vary, and this can be mathematically linked to the core/shell structure (see work by Shard). 

with  TEM, HRTEM and XPS