I just bought a bike (80's Nishiki, 3 chainrings, 6 speed cassette, Suntour Power downtube shifters, Suntour AR derailleurs). At first I thought there was something wrong with the derailleurs or shifters: the shifters are weird, they are friction but they feel like they have a ratcheting mechanism inside.
Here are the problems:
1. Shifting from big to small is fine, front or back. On the front, its very difficult to shift up to a bigger ring, and going from the smallest one usually skips the middle one straight to the big one. Either way I need to push the shifter all the way to get it to shift.
2. On the rear, going to a bigger cog I need to overshift (push too far) and then let the cable back out a tiny bit.
I think I may just need a new chain, it seems really loose/wobbly, and I measured it and each link was slightly more than one inch.
If you can discern a length above one inch for an individual link then your chain is almost definitely extremely worn... the usual measuring strategy is to measure 12 links centre-of-pin to centre-of-pin, and if it is longer than 12-1/16th inch, the chain should be replaced.
However, if the chain is excessively worn, then the cogas and possibly the chainrings are also worn to match, and a new chain will not mech properly in the old gear teeth... on modern 'hyperdrive' ramped and pinned shaped cassette teeth, a new chain on a worn cassette will skip under pedaling forces. Perhaps an older non-hyperglide freewheel will hold the chain better, but probably still not work too well. You will likely need a new chain and freewheel, and possibly new chainrings.
Whether or not this solves your shifting dilema is unclear - older bikes without indexed shifting or ramped and pinned chainrings were not so smooth shifting, and learning to shift correclty each time without missing a shift or overshifting was one of the skills you used to need to consider yourslef a competant cyclist back in 'the day.'