All above have excellent suggestions and should do.
What has worked for me is spraying the shifter with A LOT of WD 40 and working the gears on a bike stand. I put a towel down on the ground to catch the WD 40.
The shifters probably have not been used in some time.
I do this about 3 times and has worked for me. I have also put in new cables and housing when needed which is about 40 % of the time
make sure you lube after the WD 40 flush with some light grease