Get a volt meter, a $10 volt meter will save you alot of time and money.
Newer alternators have built-in voltage regulator.
There will be 1 larger wire(maybe 2) hooked to the back of the alternator, that is called the "B+" terminal.
B+ is hooked directly to the battery +, but thru a fuse
Set Volt Meter to 20vDC
Hook up Voltage meter, Black wire to battery - and then Red wire to B+, you should see 12.5volts(12.2-12.8volts)
If not then that fuse is blown or wire is hooked up wrong, wiring depends on the year.
The voltage regulator in the alternator has 3 wires on a separate connector.
Usually a white wire, a yellow wire and a light green wire
Unplug this connector
Same as above Black wire on battery negative(-) Red wire to Yellow wire in connector, should read battery voltage 12.2-12.8volts.
If not the fuse is blown or wiring is wrong.
Now the ON/OFF switch for the alternator, yes alternators have an ON/OFF switch

.
The Green wire
Same as above Black to battery -, Red to Green wire, should be 0 volts with key off
With key on it should be 12.2-12.8volts, voltage here turns ON the alternator, this 12volts comes from the key switch and thru the Volt Meter on dash and/or Battery Light on dash.
Battery voltage.
New battery will have 12.8volts, holds 100% charge
4-5year old battery 12.3volts, holds 70% charge
12.2 holds 60% charge and it is time to shop for a battery sale, next cold morning and you can get the "click, click, click" of a dieing battery.
When alternator is working:
After starting engine battery voltage should go up from 12.5volts to 14.5volts or higher, but under 15volts.
This is the voltage regulator recharging the battery after starter motor drained it.
After a few minutes the voltage should drop to 13.6volts, this is maintenance charge, it keeps battery at full charge but won't "cook it" like 14+ volts would.
It should stay at 13.4-13.8 volts no matter what the RPM or load, i.e. lights on heater fan on, etc....
That's the voltage regulator working.
If voltage drops below 13.4volts at idle with lights and heater fan on, i.e. dimming head lights at idle, that means 1 of the 3 Field Coils in the alternator has failed, not uncommon on older alternators, but since it will keep battery charged while driving(higher RPMs) you just need to keep that in mind and shop for an alternator sale, no urgency, but just a heads up that 1/3 of alternator is not working.