I cannot get the front gear shifter to go through all three of it's positions. I have adjusted the cable tension and the limit screws.

Also, the chain always rubs against the front derailleur when in one of the two gears that I can get it into at any time.

Has it always behaved this way or did it suddenly appear?

Is the problem that the lever won't pull the cable? Or that the lever won't release the cable? O\r if the cable is pulled along it's bare section the der won't move? Or that the der won't move far enough? Many possibilities but without more description of what does and doesn't happen we can only shot in the dark. Andy

Front derailleur adjustment is tough because there are many things that affect how it works and they all play off one another. If the derailleur has been bent (due to crashing, chain jamming, or something else) it can be nearly impossible to get it right.

First, start here:
https://www.parktool.com/blog/repair...eur-adjustment
Go through all the steps in sequence. Skipping a step because "I know it's right" often results in frustration.

Is there a bike repair shop near you? Many times an experienced mechanic can spot issues you've overlooked.