Ok here are some standard basic vehicle checks:
Basic car checks can be summed up by the acronym POWER:
P: Petrol (fuel including diesel or LPG), a full tank or at least twice what you will need for your journey.
O: Oils (including power steering and transmission) engine oil should be at least halfway to the top of the dipstick – other oils / fluids will have levels in your manual. A little 3 in 1 oil or WD40 in the locks will help prevent them freezing you out of the car.
W: Water (screen wash and coolant) a good winter mix for screen wash can be made of screen wash concentrate, a shot of washing up liquid and 200ml of methylated spirit per 5 litres of water (most cars have a 4.5 or 5 litre tank). Coolant should already have the right amount of anti freeze, but can be checked with a simple kit.
E: Electrics (battery, bulbs heated screen etc) is your battery a sealed unit in which case there is little to check many have a green ‘OK’ dot in a window if the electrolyte is down or the battery is near the end of its life it will show red. Old style batteries may need a top up charge in very cold weather and a top up of the water with distilled water (also known as deionised water).
Bulbs inside and out should be checked and make sure lenses are clean before setting off (a clean lens picks up less dirt that an already dirty lens), check the lights inside the car including the dashboard lights – when you turn on the electrics most cars illuminate all the warning lights on the dash as a self test. Check that your heater & demister fans work as well.
R: Rubber (wiper blades and tyres) Check that wiper blades are in good condition and not cracking, torn, split or with the edges curled over as this will reduce their effectiveness, a quick wipe with a rag soaked in methylated spirit will remove a lot of muck and road film that can build up.
Tyres need to be at the right pressures, the days of summer and winter pressures on tyres no longer really applies to most road tyres, the main exceptions are specific off road tyres, snow and winter tyres (the latter rarely used in the UK), check the sidewalls and tread for damage, impacted stones or bits of glass. Likewise check the tread for depth, the UK standard is 1.6mm across 2/3 of the tyre, however, it is recommended that tyres should be changed at around 3mm tread depth as the tyre will have lost 70% of it’s grip. Uneven tread wear can indicate problems with inflation and tracking (Uneven tread wear can also indicate problems with wear in the steering mechanism).
As for the handbrake, the mot standard is that the handbrake should fully engage with 3-5 clicks on the ratchet dependant on type and age of vehicle (according to a friendly MoT tester older cars had a longer action on the handbrake).
Footbrake - even with the engine off (though this is best done with the engine on) you should feel the pressure on the brakes steadily build up - it should feel firm if the brakes feel 'spongy' or release pressure then start to build up again it can indicate either water in the brake fluid or a small leak in the system - either way do not drive and get it checked out.
Wheel nuts - they cannot be checked using the mark one eyeball alone - some commercial vehicles use indicators - they are set on the nuts with a pair facing each other if they do not align then it can indicate that they may be coming lose, garages use (or should) proper torque wrenches for setting wheel nuts as per manufacturers specs. First check I make is are they hand tight then use a wheel brace (the extending ones are pretty good) if you can put your foot on the end and the nut doesn't turn when you are pushing down with your body weight it's pretty tight (not as precise as a torque wrench but a good field guide).
For your fluids - check the manual this should give you the location of reservoirs, filling points, fluid type and grades needed.
Hope that's of use, if you need any more info please yell.
It does really bug me that so many companies that require employees to drive all trot out the same rubbish about performing daily/weekly/monthly checks etc yet despite it not being taught to learner drivers nor is it part of the test expect company drivers to know what these checks are and how to perform them.
Sorry captainkidd that wasn't aimed at you, just a rant over a pet peeve of mine at employers who seem to view driver safety as 'outside' of 'normal' H&S often (IMO) because they haven't a clue themselves.