Since you are fairly new to car repairs I presume, one caution. Be sure not to try to run the engine with the battery disconnected. Some first time diy’er try this in a attempt to figure out what’s wrong with their battery or charging system, but the alternator must always be connected to a battery when the engine is running. Otherwise expensive electronics in the car’s computer system & elsewhere can be destroyed.
Shops can easily determine if a battery is good or not. They’ll charge it up, then do something called a “load test”. No need to wonder. And the test isn’t very expensive. Some will do it gratis, as a courtesy in fact. Best way to do that is to remove the battery from the car first. That way you know the car isn’t affecting the test.
A diy’er can get a pretty good estimate whether the alternator is working or not just by measuring the voltage (at the battery posts), comparing when the engine is running vs stopped. You’ll need an inexpensive DVM to do that. Engine stopped, battery mostly charged, it should measure around 12.5 volts. Running, if the alternator is working, around 14-15 volts. At least right after first starting a cold engine that has been sitting overnight. The voltage will usually go down to around 13 volts after the car has been idling a while.