Trouble Shooting a Problem?

Okay, so I just bought a new 2001 Grand Prix. I am not completely car savy, but I know a few things for a lady. Still, need some help. Yes, I will be going to mechanic in the end, but wondered if you guys might have an idea.

The battery won't hold a charge.

and the computer diagnostic tester thing says the Head Lamp Relay is malfunctioning.

Ideas on why the battery keeps dieing?
Some say its the relay, some say bad battery or the alternator.

Most autopart stores, autozone, advanced etc will do free battery load testing and alternator testing. Is it losing it's charge overnight? Does your battery light/charging light illuminate when you turn the key on, then go off once it it running?

Do you know how to use a multimeter?