Troubles replacing an old outlet with a GFCI outlet
Any GFCI can protect downstream parts of the circuit.  You move that circuit (hot and neutral) to the LOAD side of the GFCI.  

Here's the trick:  You must move hot and neutral together as a matched set.  If you don't, you'll get constant GFCI trips.  This can get complicated with light switches, so you have to slow down and learn what you're doing. 



Whenever you get a complicated one, seriously consider going to the previous location from which that location is fed... and put the GFCI there instead.  If that is the first location, replace the breaker with a GFCI breaker and call it done. 



If you don't see a light to be switched, it's possible that switch works a garbage disposal.  Switches near sinks usually do that.  If the disposal has a receptacle, it'll be under your sink.  That definitely needs GFCI protection.