Front gear issues with middle ring (not always making it over)
On the shifter handle, get the adjustment nut all the way in(where cable connects to the handle). Now get yourself some new brake outer and inner cable. Lube it very well, with oil(not grease, nor WD-40)

Now for derailleur, one screw is for the lower limit, the other one is for the upper limit. This is to ensure that the chain will not go out of the gears

Now shift into the lower gear, and fix the cable on the derailleur as tight as you can but without moving the derailleur. Once this is done you just need to use the adjustment nuts on the handle or the derailleur. I like adjusting it from the derailleur as the first fine tunning and any other from the handle, so I can keep track of how much wear the cables have.

Also check out if the derailleur is straight and parallel to the chain rings.  If it's not, it will not shift properly.