There are six main points, which need to be checked. This normally takes a minute or to with each bike and called an (M) check.
Both front and back need to work. Slack brakes are probably all right in the dry, but in rain the brakes should bite quickly when the levers are pulled, not when they are tight against the handlebars. Brake blocks also need to be checked for wear and alignment.
Wheels & Tyres
Tyres wear. The back tyre wears more quickly than the front as it carries more of the cyclist weight. Bad worn tyres must be replaced.
Tyres should be well inflated so that they are hard to the touch.
Wheels rims and spokes should also be checked for bumps and any other damage.
The Chain should be well oiled and have no serious defects. On non-derailleur bikes they are often too slack. When properly adjusted it should be possible to move no more then width of a pencil.
One area that is often overlooked on a maintenance check is the steering. This can be checked by standing with the front wheel held firmly between your legs while facing the saddle and turning the handle bars against the pressure from your legs, if the steering turns easily, independently from the wheel then the head bolt needs tightening.
Ideally, when sat on a bike a novice cyclist should be just able to touch the ground with at least one foot. Cycles are more likely too be too big than too small but usually this is more of an inconvenience than a serious safety hazard.
Whilst we mostly ride our bikes in daylight we should make sure that we have a red rear reflector fitted to the rear of the bike and also make sure that you have a red rear and white front light fitted to your bike if you are intending on cycling in the dark.