I own an old bike, with the old style gear shifters (levers). I have been trying to get the cable tension right. However, it seems to be getting worse. Whenever I am on a slight incline or just in a high gear on flat land the gears drop (involuntarily). Is there a certain way to fix this problem? Could it be anything besides for my cable tension?
Others have commented this and it is the likely cause.
With friction shifters the position of the lever is held by the screw that holds the lever in. When pressure is exerted on the drivetrain the chain tries to drop down the cassette, only being prevented from doing so by the rear deraileur which in turn is held by the cable tension of its shifter. If the shifter screw has loosened a bit the force from the chain will overcome the deraileur, pulling on the shifter cable, and shift down by itself.
It's simply a case of tightening the shifter screw to a degree that it's still comfortable to shift but won't be pulled out of position. It should feel pretty stiff when you shift but not so much that have to put much effort in.
On my old peugeot racer I found myself tightening the screws every month or so.
I would try to change shifting cable(s); both the inner cable (wire) and the outer casing. Just a tiny amount of friction here will mess up gearing - and that might very well be the case if your bike is old and has not been maintained or serviced for a while. Use a proper cable cutter if you need to cut the outer casing and round the hole after cutting so it is a perfect circle.