The derailleur will move just as far as the shifter tells it to. But the shifting system *is* a system, and it all works together. If your cable outer housing has a kink in it, it makes it harder to shift. If your chain is worn or your cogset is worn, that can lead to skipping. If your rear hub is out of adjustment, that can lead to "ghost shifting." If your shifters are getting worn, that can be a problem too.
New cables are a way to eliminate one variable, but be sure to get 'shift' housings and not 'brake' housings. Shift housing is designed to not compress too much over time, which brake housing is known to do.
So turn the bike upside down and look at the rear derailleur. The pulley closest to the cogs, the 'jockey' wheel (the other is the 'idler') should be directly in line with the cog above it. If it isn't, adjust the cable tension until it is. If your hanger is bent, it should be immediately apparent; tweak it the way it needs to go with a Crescent wrench closed tightly around it.
There's nothing terribly complicated about bikes. Mostly it's a matter of knowing a little basic theory, and using your head.
Start by making sure your limits are correct as stated above, the limit screws are on the top of the rear derailleur first. Once you have the outer cog in line with the derailleur or high limit, set the lower limit to the smallest cog or low limit. You can then adjust the flow of the chain with the tension adjustment at the barrel adjuster near the cable derrailleur link at the bottom of the chain stay. If none of these help, you can check your chain with a cheap chain checker tool from park or spin doctor and see if you need a new chain. Also inspect your rear cassette to make sure you don't need to replace that also. You may have to change your cassette with the new chain. This is a major fix, and you may be better off taking it to a bike shop.