Old car daily driver engine won't crank battery diagnosis

Before throwing expensive parts at this 2002 Mercury Grand Marquis, let's systematically try a couple of things first. How are the physical connections between the battery and the clamps on the ends of the cable? Does the battery just need a recharge (because I accidentally bumped the dome light on)? I used a 10mm deep-well 3/8"-drive socket and ratchet, a couple of must-own battery connection scrubbers, and an automotive battery charger.