If the SunRace grip shifter is like the Shimano RevoShift, lubrication won't help and might cause problems. They're very plasticky, essentially self-lubricating, with only a couple of metal springs or stiffeners. The RevoShift works fine until it breaks. Then the only recourse is to replace it. They aren't user serviceable. But for only $10-$15, including cables, it's worth buying a backup kit.
Check for a kink or fraying cable inside the shifters. Happens more often than I'd like to admit when I use cheap cables, like SunLite or other generic stuff rather than Shimano or Jagwire.
Double check the cable routing and clamps. With some front and rear derailleurs this can be really tricky. Some have two or more options and it's easy to get it wrong while it looks right and sorta functions.