I don't really know what you know, but I'm bored so I'm just going to start typing. If it's useful, then good... otherwise... whatever.
Integrated Shifters (I call them Brifters: Brake + Shifters)
My first guess is that there's a problem with the cable and not the shifters themselves. First try and figure out if the cable can at least move freely within the housing and replace the cable/housing if needed.
Here's a little video which should give you an idea of what things look like:
As you can see from the video preview image, the first step would be to pull back on the rubber brake hood folding it back on itself. Be careful so as to not rip it. You shouldn't need to completely remove the brake hoods. Remove your bar tape to expose the cables/housing.
For most rear derailleurs:
When downshifting, shifting from a small cog to a larger one, the cable pulls the derailleur. (Tension is increased)
When upshifting, shifting from a large cog to a smaller one, the derailleur pulls the cable. (Tension is decreased)
On the rear derailleur itself there are two screws. One is labelled with an L, the other with an H. Assuming they were perfect before replacing the cable, they should be perfect even after replacing the cable. Knowledge is power, though, so here are the rules they follow:
L=Low limit screw adjusts how far 'into the spokes' the derailleur is allowed to move. Screwing it inwards restricts the 'into the spokes' movement regardless of how tight the cable is.
H=High limit screw adjusts how far the derailleur moves beyond the small cog. Screwing it inwards limits it from moving beyond the small cog regardless of how loose the cable is.
You will need to readjust the cable tension, however. Set the shifter so it's in its highest position. (If you have an N speed, it should be in N). Pull the cable somewhat snug and anchor it to the rear derailleur.
At the rear derailleur there is a barrel adjuster which is right at the end of the cable housing. The housing ferrule actually slips inside the barrel adjuster. Screwing the adjuster counter-clockwise will tighten the cable tension. Screwing the adjuster clockwise will loosen the cable tension.
Ok, so how do you know what tension it's supposed to be?
Upshift the shifter until it is at its highest gear, then downshift one gear. The shifter should be in gear N-1. Adjust cable tension with the barrel adjuster until the chain changes gears from the smallest cog to the second smallest cog. Once shifting into that one gear is set, shifting into all of the others should also be set. Other than possibly needing to fine tune it one 'click' at a time, it should be set.
Run through the gears to ensure that the derailleur will not shift beyond the 1st or Nth gears. Fine tune the L and H screws if needed.
If your problem is with the Integrated shifter (brifter) itself, sorry I can't help. I have no experience with them; I've got bar-end shifters on my road bike.
Hope that helps.
If not, whatever.