Originally Posted by Russ Roth

How old are the actual cables and housing? Although it can happen with any shimano group my experience is that 105 is the worst at having the cable fray in the housing which shows up as lazy shifting. It also doesn't take much for a cable to develop a small kink that isn't as noticeable under tension but can still make for lazy shifting. If the chain is worn, have you checked for chain stretch? The wear can show up in side to side movement and the chain be a little too floppy to easily make the shift.
If I know I have good cables, housing and chain. Then first is to align the front der which those videos should have covered, basically outer plate parallel to the outer chainring, height (which you should avoid messing with) just a couple dimes width above the chainring. I like to pedal the bike which the rear der in 1st/lowest gear and push the front der cable down with my thumb where its heading down the downtube and pull it away from the frame so the der shifts to the large ring, then I slowly let the cable retract till the der drops the chain to the small ring. I might have to do this a couple times but it tells me where the lower limit screw is adjusted to, once I set the screw I do it a few more times quicker to mimic the shifter and make sure it drops with ease. If it continues to shift smoothly I tighten the limit screw in 1/4 turns till it isn't quite smooth and back off a 1/4 turn. Once that's set, make sure your adjustment barrels are 90% screwed down and retighten the cable at the fixing bolt. A 3rd hand cable tool helps to get the tension tight. Now with the upper limit screw backed out, shift to the large ring and tighten the limit screw till the front der just moves towards the inside. Move the rear der to your high gear, drop the chain the small ring and using your thumb again while pedaling the bike, pull down hard on the cable. The goal here is to see if the chain will overshift off the crank, if it does tighten another 1/4 turn and try several more times. Once the chain doesn't overshift, and should be easily dropping to the small ring you can adjust the cable. If you had to tighten the upper limit screw a lot to stop the chain popping up from the shift that told you where to set the upper limit screw to begin with you might need to let out a little slack with the adjustment barrel. Otherwise, just tighten it a couple turns keeping an eye that the der doesn't move off the lower limit screw from chain tension and you should be good. With a good cable, housing and chain this has never failed me for setting the adjustment.