Help needed tweaking the shifters

Assuming your front derailleur is really a triple, and not a double, you most likely need to (re)set the hi and low stop screws. Releasing the tension on one or both limit screws will let the parallelogram shift farther inward and/or outward. Refer to the Park Tools website for directions if needed.

On the assumption that you have Shimano brifters and FD, you should be able to download the installation data sheet for your parts, from Shimano's web site (which covers initial adjustments, also).

It's also possible you have these tucked away in the paperwork that came with your bike.

Actually, once your FD is set up you shouldn't need to tweak or adjust anything. My bike doesn't have any barrel adjuster on the FD at all. It hasn't needed any adjustment since the initial break-in period 24K miles ago.

If you want to do it yourself, agree the Parktool website is a good one. Would be easier of course just to take it back to your LBS: they should be able to diagnose the problem. If it's simply a cable adjustment, they can do that quickly for a small labor charge. But with your description of the issue, I'd guess something else might be the problem.

The simplest way I know of adjusting front derailleurs:

You need a means to lift the bike off the ground (a trainer or a bike stand), some allen keys and a small screw driver

1) Shift into the biggest sproket and the smallest chainring (first gear).
2) Undo the cable and get it out of the way
3) Loosen the clamp and ensure that the cage will have no more than 1.5mm cleanarce above the big ring. Also ensure that the inside plate of the cage is parallel with the chainrings. Re-tighten the clamp, ensuring that you don't change this alignment in the process.
4) Screw in the lower limit screw (L screw) until the inner plate rubs on the chain.
5) Install the end of the cable under the clamping bolt. Manufacturers provide a groove here for the cable to sit in. The cable MUST sit under the calmping bolt in the correct orientation or the derailleur will not work correctly. DO NOT wrap the cable around the bolt. Pull the cable with your fingers and tighten the bolt.
6) Now unscrew the lower limit screw (L screw). This will take the derailleur back to its correct starting position, and , at the same time, will add the correct tension to the cable. The chain should not rub any longer if adjusted correctly.
7) Shift to the middle sprocket at the rear and then shift back and forth between the small and middle rings. If the shift takes the chain too far and causes rubbing, suspect that steps 4-6 have not been done correctly, and repeat with greater care. If the shifting is too slow then add more tension with the barrel adjuster at the shifter.
8) Now shift to the smallest sprocket and attempt to shift between the middle and big rings. Now you must adjust the H screw so that the derailleur can shift to the big ring without rubbing, and also so that the derailleur will not throw the chain off to the outside.

Once this is done you should test that all is good by running back and forth through all gear options. Small tweaks of cable tension or the L/H screws may be necessary but any more than 1/4 turn on the screws is an indicator that the initial alignment (steps 4 and 5) hasn't been done correctly.
Steps 4 and 5 are the most difficult to get right and it might take a few attempts, the first time round, but once you get one type of derailleur down-pat it's easy to repeat the process again later.
This is also the only way to get close to the correct cable tension. Once the cable is cut you have no hope of applying the correct tension by hand.
If this process is too complicated, then it's best to go back to the bike shop.