Back gears don't want to shift down easily

[rpmdesigns]

I just took my bike to the bike shop where I got the bike and had my second free tune-up.
Really I was having terrible shifting problems so I took the opportunity and used the tune up. Turns out the rear derailleur cable was stuck so they replaced it.
So, now I'm having trouble again (although not as bad). I should mention that gear and brake adjustments are free for life for my bike at the same shop-- but I'd rather do something myself if it is quick and easy (and free or cheap) instead of taking the time to put the rack on my car, getting the bike to the shop, and waiting a few days to get it back.
My problem is that example, I am riding on 5th gear (back) and want to shift to 4th. My bike has revo shifters that I rotate forwards or backwards by gripping them and twisting my wrist forward or back. So I twist my shifter to go to 4th, and I have to hold the shifter for what seems like forever (5-10 seconds max), and then I get the shift to 4th.
This happens when I shift from most gears except maybe from 2nd to 1st.
Also, sometimes I will shift UP, and I get a shift to 2 gears up. Example: I want to go from 3rd to fourth, and I end up in 5th. I can tell from the resistance. I look down into the gears and I can clearly see that I'm on 5th. I then have to again twist the shifter and hold it until I go to 4th.
I looked at the video on this site on how to adjust the rear derailleur. Am I right in thinking that all I need to do is to adjust the tension on the cable?
Any comments and/or questions would be greatly appreciated. =)

[DaveM]

It's common for a new cable to stretch a little. Try tightening it up a little with the barrel adjuster where the cable goes into the rear derailleur.

[rpmdesigns]

That does makes sense... I can understand that a cable will stretch shortly after installing it.
I think I've made some progress.
After my morning ride this morning I decided to give the barrel adjuster a try because the chain fell next to the 7th gear (outside of it) in back and I had to stop and fix it.
I forgot from the video which was tighten and which was loosen. I tried one direction (away from me as I looked at the barrel from the side of the bike) and took a quick ride in front of my house and it seemed to be doing better.
Then I went on my Friday ride to work and after a while I noticed that once again, down shifting in back was requiring me to hold the shifter while pedaling until I got the shift. I stopped at a couple of intersections where I needed to wait anyway, and got off the bike and messed with the barrel adjuster again in the same direction. The first time it got better and then quickly got worse again.
The second time I was almost to my destination. I rotated the barrel even more times and I could tell that it was getting tighter.
Shifting felt much better for the rest of the ride. We'll see how it does during my ride back home this evening. I will report back and we'll see if I need more help.
After the initial test I feel much more comfortable messing with the barrel adjuster. Later I will try to get comfortable with the other screws that were mentioned in the video.

[rpmdesigns]

No problems after making this adjustment. I even had to re-adjust in the same way when the shifting started to delay a little again. Shifting smoothly again.