>>1945441>>1945397now use PB Blaster or something marketed as penetrating oil with a tiny straw and blast it into both ends of the shift cable housing while working the shifter back and forth. your cables and housing may be beyond repair with rust but you won't know until you service the shifters. if your shifters are smooth but the cables don't move freely, replace cables and housing.
if the shifters aren't smooth, they may free up with some penetrating oil . if after doing the cable housing the shifting moves smoothly, then you can skip it , but put some lube on the pivots for preventative maintenance. if it isn't freely moving, unbolt the cable, PB blast inside the shifter and work it. if you have trigger shifting, the little ratchets and pawls can be gummed up with old grease, you would need to unbolt the cover and get in there with solvent, a scraping edge, and a wire brush. very tedious . then lube .
now rebolt the cable: turn shifters to lowest gear, derailleurs should be in line with the inboard-est cogs. if not, there are limit screws on the derailleurs, turn the low ones until the cages line up with lowest cogs. now reclamp the cables while holding them so there's no slack but also no extra tension while bolting.
put the chain back on, needle-drop lube into each roller then wipe down the exterior.
you're drivetrain is now fully serviced, you've eliminated all problems related to it. if it still gives you problems, do the aforementioned barrel adjustment now.
at this point, you *should* be golden. the only other problems could be :
1) the FD is clamped at the wrong height/angle.
2) the RD hanger is bent. if it's steel you can carefully bend it back with e.g. an adjustable wrench, if it's a bolt-on hanger you can replace it.
3) the derailleurs are so worn from use they're loose and blown out which is extremely unlikely but you'd have to replace them.
that should be every