You need to twiddle the adjuster on the shifter like you say.
The best way to learn is to pedal and alter the adjuster and see and hear what happens, that way you know what is going on.
Then once you've figured it, go for a ride and do it then.
While riding
Start with shifting down, if it doesn't go down after a click then release some cable (turn the adjusting so it goes in) until it does. Once it drops down add say a 1/4 twist.
Then do it as it goes back up to the larger cog one by one. If your lucky it'll be perfect

if not and it will not go up then a drop to the smallest again and add a little turn (barrel adjust out, shouldn't need much ad you've got just over a 1/4 turn to play with really) keep trying till it feels right.
This way you get an idea of what is happening and should be able to hear the problem in future so you can alter on the move if needed.
Shadow rear mechs are also (as I've found out reading Shimano docs) more critical the the jocky height, (i.e. adjusting b-tension screw).
They give quite a specific distance it should be, ... (EDIT found the link now
http://bike.shimano.com/publish/content ... le.html/14)%20Shadow%20RD%20Tips.pdf)
I'll add, while I've not used this I came across this page while helping someone else
http://www.exchangecycletours.org/rdadjustment.htm