Things every driver should be able to do...
- Pump their own gas.
- Pop the hood & identify basic areas/parts under it.
- Check & top up oil(s), coolant, washer fluid.
- Change wiper blades.
- Check and adjust tire pressure.
- Jump-Start a drained battery.
- Check tires for wear.
- Install spare tire.
- Check/Tighten lug nuts.
- Notice unusual noises/behavior & under what circumstances.
^^ They should make those a mandatory part of getting a license, even if you never use the knowledge.
Most of the jobs you should be able to do yourself were covered in Xist's post...though those things with potentially disastrous results (Oil change, brake work) should not be done by someone truly...incompetent. At least find someone to show you how and check over your work the first few times.