First, check to make sure your rear wheel is fully within and centered in the dropouts. With the chain on the front middle ring, check shifting of the rear drivetrain. A bike should have the rear shifting tuned first before moving on to the front. If shifting in the rear is satisfactory, leave the chain on the largest rear sprocket and shift the front to the small chainring.
I wonder if some of your symptoms can be explained by excess friction in the cabling. Check your cables making sure the outers have nice looping curves and no sharp turns. Check that the ferrules are in good shape and sit fully within their stops, if any. Examine the exposed portion of inner cable especially the sections that move in and out of the external housing around the stops. Any fraying or kinks could impair shift performance. The holes in the ferrules tend to wear such that sharp edges protrude from around the hole and can snag the inner cable as it passes during a shift.
A good way to test the inner cable movement is to disconnect from the derailleur pinch bolt, hold the cable firmly with your hand and throw the shift lever, then release (like your shifting to the smaller chain wheels). The movement should be smooth without any resistance or catches.
While you're on the small ring up front and large cassette cog in back, and while the inner cable is disconnected, see where the outer derailleur cage sits in relation to the large chain wheel. The beginning position of the outer derailleur cage should be parallel to the teeth of the large chainring. You can take the long side of an Allen wrench, lay it flat against the outer part of the large chainring with a portion of the wrench sticking up toward the front derailleur. Move the crank while you hold the wrench flat. The proper starting position of the front derailleur is for its outer cage to be parallel and in the same plane of the outer aspect of the large chainwheel. The Allen wrench is a guide. It should just brush past the outer cage of the derailleur without being deflected by it, nor should there be a gap between the wrench and derailleur cage. This technique checks the derailleur alignment: parallelism with the chainring and the correct starting position of the derailleur (outer cage in line with the large chain ring). Manipulate the derailleur low limit screw to get this starting position. Then turn the cranks and see where the chain runs in relation to the inner cage. It should be close but not make noise from rubbing the cage. I usually turn the L limit screw until the chain starts to rub then back the opposite way until the noise just stops.
One factor that can mess up front shifting is routing the inner cable incorrectly. Especially near the pinch bolt. It's easy to miss going over a leverage tab at the pinch bolt. So make sure the inner cable is routed correctly. Be careful not to tension the cable so much initially that it moves the derailleur cage before it's secured in the pinch bolt.
Next step is setting the H limit. The next step after the H limit is at the middle chainring to set cable tension. The chain should be on the large cassette cog again. An important point of this step is that when setting the cable tension here, you want to have reached the middle chainring by shifting down from the large ring (as opposed to up from the small ring). With the chain on the largest cog and middle ring in front (by shifting down from the large ring), adjust the barrell adjuster so that the chain runs very close to the inner cage plate without touching. Here again, I usually will manipulate the barrel adjuster so the chain rubs the inner cage plate then turn the adjuster the opposite direction until the noise just stops.
