I was out on a ride yesterday and several times when I was changing up and down between big chain ring and little chain ring (I only have the 2 rings up front) it required 2 gear changes. The gears actually settled on an imaginary middle cog! How can I fix this?
I think Shimano integrated shifters ("brifters") for the front derailleur on a triple normally have 5 indexed positions: 1-3-5 are the main positions that match the chainrings and 2-4 are intermediate spots to avoid chain rub for some chainring+sprocket combinations. If you give the front shift lever a short pull it will click once and the front derailleur moves one position. Give it a full pull and it should click twice, moving two positions.
Normally going up to a bigger ring requires a 2-click upshift; sometimes you need to give it a one-click downshift afterwards to eliminate chain rub if you're in a larger rear cog. Going down a ring may only take a 1-click downshift, depending on what position the derailleur is in. All of this is based on my experience with a triple, but should work the same for a double.
If shifting problems are more noticable going up to a bigger ring (needs more clicks than before) then this is probably due to cable stretch. If you have a barrel adjuster on the front derailleur cable, try turning it counterclockwise (looking at it from the side where the cable housing enters it) to tension the cable a bit.
If the problem is when dropping down to a smaller ring, then either the cables are sticking somewhere (worn cables/housing) or the front derailleur is sticking (needs lubrication or has a worn spring).
This is probably down to the gear cables.
If the cables are new they may have stretched a little and you'll need to take the slack out by adjusting the tension, you usually do this with a barrel adjuster either at the shifter or inline in the shifter cable.
If the cables have been on the bike for some time then the problem is probably sticky cables, water and dirt get inside the cable outers and stop the shifter pulling enough cable through. from your description this seems the most likely option. You can try cleaning the cables but in my experience new cable inner and outers is the way to go.
It is also possible that you have extra positions on the shifters to accommodate the chainline. I know my Sora group on my Trek has two lower positions. If I'm in the lowest position then my chain will rub slightly on the smallest cassette gear. I'll move it to the second position which doesn't shift it but removes the rub on the front derailleur.
When I shift up to the large chain ring from position two is goes fine. When shifting down I need to shift two positions to get it onto the small chain ring. The shifter is set for this and a full throw on it is two positions.