Can I simply choose one shifter without taking into consideration the derailleur that will be used with it and vice-versa?
Derailers differ in the amount of "pull" needed to get from one gear to the next, and, for front derailers, whether they are "spring in" or "spring out". Neither of these mattered much before indexing, but with indexing a mismatched shifter will mean that you can never get shifting to work cleanly over the entire cog range.
As imel96 says, though, generally a shifter of the same brand and number of gears should be compatible with a given derailer.
If you don't already have something in mind, first find a shifter you like, and it should be easy to go from there because whoever you buy the shifter from should be able to tell you which derailleur is going to work with it. Or vice versa, start with the derailleur and then there will be a specific set of compatible shifters.