I also need to get the derailleur cable adjusted correctly. I don't know why, but I never can get those set up to work 100% like they should. My method has always been to put the shifter in the highest gear (7) and to make sure the chain is on the smallest cog. Then I make sure the cable tension is tight and tighten down the cable nut. Then I do any needed fine tuning on the little cable adjuster up on the shifter. Did all of this last night and it didn't shift right so back to the drawing board.
The 2 screws are limit screws. They have nothing to do with the shifting on indexed systems. the one limit screw keeps the der out of the spokes, the other limit screw keeps the chain from jaming between the small gear and frame. Once set at the factory, they should never need adjusting unless you crashed on the der and bent it or the frame. The only other time they need adjusting is when some clueless twit has been messing with them.
To adjust index systems:
Shift to a middle gear (4th in this case of 7 gears), make sure the grip shifter is in 4.
Adjust the cable housing barrel adjuster at the rear der until the chain runs directly straight down from the 4th cog to the top jockey wheel. You have to look at this from behind the bike.
The chain should run smooth at this point. Any jumping and you need to fine tune the adjuster.
While you are back there, look to see if the cog, top jockey wheel and bottom jockey wheel are in the same plane (ie, frame/der hanger or der are not bent)
on grip shift shifters, there is a second barrel adjuster on the shifter itself that does the same thing. That allows you to fine tune while riding.
You will find that more gears you have the more touchy the adjustment. 10 & 11 cog cassettes are the most finicky. 7 speed systems are typically pretty stable.
On indexed gear systems, the index (click) is in the shifter and the der is held in place via a long skinny cable that stretches over time as it's under constant spring tension. Cheap cable housing gets compressed at the same time. Some brands are must more stable. keeping the cables clean and rust free helps shifting. Keeping the insides of the cable housing rust & gunk free really helps too.
Brake cable housing is spiral wound with a plastic cover and some times a plastic lining. It's relatively soft and flexible and doesn't work well for indexed shifters. Shimano makes special housing for index systems where the housing is made of steel wires run lengthwise that are embeded in the plastic. This type can't compress or expand so shifting usually works better.