It sounds like everything is OK. The car starts, right? Is your battery one of those "maintenance free" units that has the little green indicator? If so, what color is the indicator? Green-Good, Black - not so good. You can check to make sure all the cells in the battery are OK by turning the ignition to the ON position and look at the Volt Gauge. If it'* at 12 the battery is at least putting out 12 Volts. If it reads 10, then you have a bad cell and you need to replace the battery. Can you open the battery caps to check the water level? A Hydrometer is a device to test the battery acid for Specific Gravity. Of course, if you don't know how old the battery is, or if it is more than 3 years old, you might want to purchase a new one before winter sets in.
The OE Alternators are usually good for 50K. It'* the rear bearing that is prone to failure. I know mine went bad at about 65K. If you choose to replace yours BEFORE it fails, make sure you replace it with a good rebuilt Delco or from a BF Vendor like Domestic Performance.
2006 Salsa Vibe
2002 Snap Orange Beetle - Turbo
1991 Olds "Orphaned" 88 - Donated to Science!