Front derailleur adjustment should be done only after the rear cogs are indexed properly. If that is the case, it is most beneficial to tune the front derailleur systematically as Argenti notes. This begins with positioning of the front derailleur. Depending on how your bike is designed to mate with the front mech, there is varying degrees of flexibility in derailleur's positioning. Band-on front mechs have the most possible range of positioning, and in greatly decreasing flexibility of position are E-type bottom bracket mounted, the similarly, E2 mount, and then the braze on and direct mounts. I mention this because my suggestions will be limited by the type of mount. Front derailleur's are also specific to the number of chainrings--as you may know there are front double and front triple derailleur's and ideally the should be paired with the correct number of chainrings to match shift points. (It's possible to get them to work with the other "incorrect" number of rings, but that's very touchy and no way to go long term). My point in all that is make sure you're dealing with a compatible front mech for the number and size of your front chainrings.
Two important things with the placement of the front derailleur are first, alignment: looking down on your front drivetrain, the outer cage plate should be parallel to the large ring. Another check of the parallelism is to put a straight edge (the long aspect of an Allen key works good) touching the outer cage plate of the derailleur and the outside of the big ring. With a small rotation of the crank you should note the Allen key remains in contact with both the chainring and outter plate if the derailleur is aligned properly. (As Argenti notes, one can toe the cage of the derailleur out slightly, which brings the inner plate into contact with the chain sooner on a shift up to the big ring. This is possible only with a band-type mount and to a lesser degree, braze on. I suggest staying parallel for the time being. That's how a properly matched and tuned front drive system is designed to work).
The second aspect of the derailleur placement should be proximity of the outter cage plate to the teeth of the big ring. Here the closer you can get the outter plate to the tip of the chainring teeth--without touching/interfering, of course--the smoother and snappier you'll find your shifting is. Try to get the gap to between 1 and 2 mm. Three to 5 may work, but best results are found with less. Here again, the type of derailleur mount effects what's possible as the range of veritical movement is limited with E-type and direct mount systems.
The setting of the limit screws is generally the next step starting with the low, small ring. With the chain here and around the large cog in back, you set the limit screw to the point where the moving chaiin just stops rubbing the inner cage plate. My technique is to turn the crank while tightening the low limit screw until the chain just begins to rub, the I back it off 1/8 of a turn. I continue turning the crank to make sure there's no chain rub. (When setting the low limit, there should be NO cable tension on the system).
Connecting the cable at the pinch bolt (with the front shifter in low and the cables Barrell adjuster 2-3 turns out from full clockwise) keep in mind that the cable route must be exact --ran correctly through any leverage or directional changing ramps and at the pinch bolt there is often a leverage tab that the cable must go correctly around before being secured. If not routed correctly around such leverage points, shifting is hinky--especially up to the large chainwheel.
Finally, the upper limit adjustment is supposed to be untill the outter cage plate just stops rubbing on the chain when the chain is running on large front and small rear cogs. However, my experience has been that the high limit screw may need to be loosened a bit more than that to ease the shifting to the big ring.
Here's what I could've said much more succinctly regarding your quandary: you may find that checking the cable routing and correcting any errors there will solve your problem, and if youre certain of the correct routing, turn your front mech's high limit screw anti-clockwise a half to a whole turn and see if the shift to the large ring comes about.