Replaced faulty Shimano twist shifter but front derailleur still won't catch 3rd
That bike, and many similar lower-end bikes, comes with a clamp on derailleur, so the steps to troubleshoot your problem are:


Pivots do get dirty; give it a thorough cleaning with degreaser and then drop a few drops of chain lube on the pivots. While you are doing this, dial all the barrel adjusters back to their beginning positions.
Check the derailleur cage and make sure it is not bent.  
Make sure the derailleur is mounted to the frame properly. The cage should be exactly parallel to the chain rings, and it should be placed low enough on the frame that the teeth of the largest cog barely clear the bottom of the cage.
With the cable and chain loose, ensure that the derailleur cage moves freely by hand. You can take this chance to begin to adjust the limit screws. Start with the low side, making sure it easily comes to rest to properly shift to the small ring when there is no tension on the cable. At this point it should be clear whether or not the derailleur is functioning properly.
With the cable housings detached from the guides Attempt to move the derailleur by gently pulling the cable by hand near the shifters. If this works the problem is likely in your shifters, or with friction in the cable housings. If you can shift it by hand, this is a good time to start to dial in the high limit adjustment.
If you make it this far, adjust the cable tension and dial in the shifting with the barrel adjusters.


My experience with that particular bike is that the derailleur is frequently mounted high, is easily bent, and notoriously difficult to keep in adjustment.