I had a service station carry a brand of battery that was failing
after 6 months, he dropped the brand, a service station can load test
your battery and alternator on the car, it could be just dirty or
loose battery cables but who knows it could be more

you can check the output of the alternator with a volt meter, if the
alternator is working it should read 14.5 volts or a little higher. Just
connect the positive and negative on the battery with the engine running.

If you have an AutoZone or advanced auto parts they could check it for you,
for free.

Do a quick voltage check (but it may not diagnose a bad diode in a
rectifier bridge). And as others suggested, take it to AutoZone or
your local parts store that offers free testing.

    http://www.autozone.com/in_our_stores/free_testing.htm

There are a number of things to check:

1. The alternator belt tension. The belt may be loose. For this I'd
use the Gates Krik-it II gauge. It about $23 on Amazon or your local
NAPA store.
http://www.gates.com/europe/brochure.cfm?brochure=2506&location_id=2977

2. Connections you mentioned. Oxidation under tight connections can
still cause voltage drops. That's why people use $2.99 battery
terminal cleaners, among all the wire connections.
http://www.harborfreight.com/cpi/ctaf/displayitem.taf?Itemnumber=94450

3. Alternator brushes. These come in an assembly that costs ~$20 from
the dealer. You might want to check these before replacing the
alternator. Brushes wear out after 120-150K miles. The assembly is
easy to replace.

4. Alternator diode bridge/regulator. A bad rectifier bridge can't
rectify the AC voltage into DC properly. So you end up charging AND
discharging the battery in quick cycles. This will damage the battery.
If the bridge/regulator is bad, just get a new/rebuilt alternator,
it's more cost effective.

5. Well, you have a new battery so we assume it's good. Typically ones
made by AC Delco (maintenance free, ~$80) or Johnson Controls
(maintenance type, ~$45) are very good.