If the battery has not been damaged, and if the alternator and charging circuitry are working, then just driving around will indeed recharge it.

If, after driving around for fifteen minutes or so, the car can't restart itself, something is defective - maybe the battery, maybe the charging system. If it passes that test, but needs to be jump-started the next morning, the problem is almost certainly the battery.

Lead-acid batteries (this is the type you typically find in cars) will last for many years if they are kept fully charged or close to it. Standard car usage will meet the battery's needs. Starting will discharge the battery a little bit, driving will return the battery to full charge and keep it there, and letting the car sit unused for a couple/few days will do no measurable harm.

If the battery becomes deeply discharged, the car's electrical system is fully capable of putting as much power back into the battery as it will hold. There is nothing to be gained by using a standalone charger, except that it might be more convenient to plug it in overnight than to drive it for a long time.

What damages car batteries? Extended deep discharge.

If a lead-acid battery is deeply discharged, it will degrade - slowly, but irreversibly. Leaving your lights overnight won't destroy the battery, but it will reduce the battery's capacity a little bit. Doing that repeatedly, or leaving your light switch on for a week, might leave you with a battery that will no longer hold enough charge to start the car, no matter how long you drive, or how long the battery sits on a charger.

Batteries will discharge themselves very slowly even when not hooked up to anything at all. Batteries in cars will discharge slightly faster, if there are any electronics in the car that still draw power even when the key is removed from the ignition switch. So if the car is going to sit for a long period, it is a good idea to hook it up to a trickle charger to keep it from discharging. ("Battery Tender" is a popular brand, in fact so popular that many people think that it is a synonym for trickle charger.)

Sitting for long periods without a trickle charger will damage the battery, slowly but surely. f you have a stereo the forgets the radio stations when the battery is removed, then you have a stereo that draws a tiny bit of power from the battery even when you're not using the car. Some Engine Control Units (ECUs) also draw a bit of power to remember tuning parameters that they learn while the engine is running. All of this can add up to a permanently impaired battery if you let the car sit long enough without a trickle charger to keep it charged.