There are myriad videos and articles online that take one through the correct process of setting up and tuning a derailleur. The Park Tool website is a frequently referenced source for "how to" regarding bike maintenance. Take a look at their rear derailleur adjustment procedure for guidance.
One thing to note regarding your situation is that cause of the majority of poor shifting problems in a mechanical system is excess cable friction. The inner shift cable is somehow hindered in it's movement which causes the force of the shifter to not be relayed smoothly and instantly to the derailleur and thus the derailleur movement is also hindered. There are many causes of excess friction including rusty inner cables, dirt getting inside the outer housing, a worn cable ferrule that can prevent the inner cable from running cleanly through it when shifting, an overly sharp curve of the cable's outer housing, a too tight adjustment of cable tension can prevent quick, smooth shifts especially when coming down the cassette to successively smaller cogs (note that the primary adjustment when experiencing rough or slow shifting from a larger to smaller cog is clockwise rotation of the barrell adjuster which has the effect of decreasing inner cable tension). While the list is extensive for specific causes of shifting trouble, most all relate to a less than optimum movement of the inner shift cable.
Regarding your situation, I recommend starting from scratch, meaning after examination of the cable system--no evidence of excess wear or corrosion of the inner and outer cabling and ferrules and that the ferrules rest fully within their respective cable stops on the frame--disconnect the inner cable from the derailleur pinch bolt and while holding it in your hand with mild tension, operate the shifter both ways. The cable should feel as though it moves freely in response to the shifter. You'll be able to feel any resistance to the free movement fairly readily. Any roughness or delay in the cable movement will be felt at your finger tips. Next important thing to do is evaluate the derailleur alignment. From behind the bike, observe the derailleur hangar. It should be straight down--no lateral deflection or twisting. One cue I look at is the gap between the derailleur hangar and the small cassette cog. Is the gap the same width along it's length? The hangar and cassette cogs should be parallel to each other, and an installed rear derailleur's cage plates that house the pulley wheels should also run parallel to the cassette cogs when cage is held down by hand or by an installed chain. If there is obvious deviation from this cage-cassette cog parallelism, the hangar should be replaced or set to proper alignment with a special tool that all bike shops worth doing business with has.
With the inner cable disconnected and a finding of good alignment between derailleur cage and cassette cogs, you're ready to begin the derailleur setup and adjustment detailed in multiple online videos and tutorials. One final note is that generally, once you have your limit screws properly set, there is usually no reason to mess with them unless you've changed the cassette or changed to a wheel that has a bit different spacing. After the limits are set, only manipulation of the barrell adjuster should be necessary when encountering less than optimum shifting.