If you ride the bike alot and you haven't cleaned the drivetrain, you
may not be jumping cogs as much as skipping cogs. As the drivetrain
wears, the chain can actually skip over the teeth. I'd start out with a
new chain and new cassette.
In the meantime, the best way to set up the shifting is to leave the
front in the middle chainring, then set the rearr to the smallest cog.
There really is no better way to learn how to fix you bike that just
doing it.
(I hate to have to have to say this, but make sure you're turning the
pedals while shifting)
Shift up to the next cog. If it doesn't make it shift back down on give
one of the barrel adjusters one _full_ turn OUT. Then shift up again.
Repeat until the shift up is successful.
Once the up shift works, make sure it still shifts down ok. If, in the
2nd smallest cog, the chain does not fully down shift to the smallest
cog, turn the barrell adjuster one _half_ turn IN. Shift up to #2, then
back to #1 and adjust the barrel 1/4 turn (or less) either way to
accomplish a crisp shift going one up and one down. When this works,
start working your way up the gears. As you get towards the big cogs,
you'll probably find that the shift either does not complete or is
noisy after the shift. Stay in the offending gear, shifting one cog
back and forth and adjusting the barrel 1/8 turn or less either way
until the shift completes.
When the shifting works at the big cog to 2nd biggest cog, go back to
the small cogs ad start over again. If the drive train is as worn as it
sounds, you'll most likely endup with a noisy shift at one end of the
range no matter what you do (because of excessively worn sideplates and
shifting ramps). You have to comprimise and balance it out.
Once you get the new chain and cassette, following the above process
and it shift completely and smoothly no matter where you are in the
range and no matter how many cogs you shift up or down.
