The bike is like new except it has an outdated drivetrain. And it's like new except it doesn't shift. Even walmart bikes today are showing up with 21-sp drivetrains. While the bike was in storage, time moved on, and cycling components continued to evolve.
You have several options, any, all or none of which might work for you.
1. The grease in the shifter may have dried and become gunky. I am not familiar with the SG60 shifter, but most shifters can be opened to allow the innards to be degunked with a stream of WD40 followed with a light lube. In many cases, where damage or wear is not an issue, this works.
2. Shop around for a decent used or NOS replacement shifter of the same type. This may mean scouring ebay for weeks with no results. But it may also prove successful for low cost, though you would need to have the shop install it.
3. Have the shop do the job. Parts and labor aren't always cheap, and it's no one's fault that your bike isn't worth that much to you. The job they quoted may include moving up to a later (and more readily available) 7- or 8-sp setup; this would necessitate replacement of rear gears (cassette or freewheel?) and chain in addition to shifters.
4. Forget the repair, sell as is, and put your money into a new bike IF you plan to do much riding.
I have no idea what a Fila Tome is. Was that a dept store bike? If so, I would go with option #4.
