I have a few questions;
Has the bike been sitting for a longtime, unused?
Where the gears working ok the last time the bike was ridden?
Has the bike been exposed to the elements or stored inside and dry?
The front shift working ok in the up direction and hanging in the down direction sounds like a
sticky or binding cable. Will the front shift properly if you pull the cable along the downtube (like a
guitar string) instead of using the shifter?
Will the rear shift one cog at a time if you use the same method i.e flex the cable along the downtube?
If true to either question then the problem is likely in the cable run from the shifter to the downtube cable stop.