Your mention of just grinding reminds me. This happens when the cable is too tight and the high limit screw is too far out. The derailleur slams the chain into the ring with so much force that it can't make the jump, so it just sits there and grinds and tries to bend the chain ring.
Try this. First, get the chain on the smallest (highest) cog in back. From the middle ring, bowstring the derailleur up to the big ring, and then use the lever to shift to the big ring.
Observe the clearances between the chain and the side plates of the derailleur. Ideally, you want the chain to barely clear the outer plate. Loosen the cable til this happens. Loosen it a bit more, for good measure, so it rubs the chain ever so lightly. Next, tighten the limit screw so when you yank on the cable the derailleur just wiggles ever so slightly.
Now we're going to see how the derailleur lifts the chain with less movement. Still on the big ring in front, shift to a middle cog in back. Now see if you can shift between the middle and outer rings in front. If it still grinds, there's something wrong with the shifter or the cable, or perhaps the lifting humps on the inner plate are worn off. If there's simply not enough movement, add a bit of tension to the cable and let out the limit screw a smidge. If it shifts, try this move from a smaller cog in back. Repeat through to the smallest rear cog, and if the chain still rubs on the outer plate, let out a bit of limit screw and take in a bit of cable, but just enough so it barely clears.
Run through all the gears and let me know if that still doesn't do it.