First, do you have a Volt Gauge on the dash and "Battery Light", or do you have a AMP Meter on the dash, it will have + and -
You need a Volt Meter to test system further, shop should have done these tests, fast and simple
Set Meter to DC Volts, 20 or higher
Key off
Test battery voltage after vehicle has been sitting for a few hours, over night is better
12.8v is New Battery
12.5v is 3 year old battery
12.3v is 5/6 years old and time to shop for battery sale
12.2v, or lower, is drained or failing battery
Remember this voltage, exactly
Ground the meter's black probe to battery or alternator's case
Touch red probe to "B+" terminal on alternator, it has the larger wire(s) connected on a stud with nut holding it, usually on back side of alternator
Should see "Battery Voltage", the number you remembered
If not then you have a bad/blown Fusible Link(same as a fuse)
OR AMP Gauge shunt
Unplug the 3 wire connector on alternator(this is the voltage regulator wiring)
There will be a white wire, yellow wire and green wire
Put red probe on Yellow wire, should see battery voltage, exactly
If not then bad fusible link
Test green wire, should see 0 volts
Turn on the key, engine off
Test green wire again, should see battery voltage, this green wire is the ON/OFF switch for the alternator, so VERY IMPORTANT
If all wires test OK
Start engine
Test battery voltage again
Should now be 14.5-14.9volts, voltage regulator is in fast charge mode
After 5-8minutes of idling or driving without shutting engine off..............battery voltage should read 13.5-13.8volts, this is maintenance charge, keeps battery "topped up" but won't damage it, 14+ volts will "cook" a 12volt battery long term
If you don't get over 13.5volts with engine running then new alternator is bad, it happens
Assuming vehicle wires tested OK