Mechanical Checks:
Open the bonnet and check things over before starting the car. As mentioned, if it's already warm, this can be suspicious.
Some guides will say that a clean engine bay is also suspicious as it could mean leaks have been cleaned instead of fixed. While this can be true, some people (like me) keep their engine bay clean, so don't hold it against them. If the car is a bit special and/or the owner is an enthusiast, a clean engine bay is a good sign.
If you're able to see the brakes well enough through the wheels, check the condition of the brake pads and discs. The pads should have a good amount of material left on them (red line in the image) and the discs shouldn't have a large lip on the edge (blue arrow); a sign of wear. These aren't deal breakers but can be used as points to negotiate on the price.
Look up inside the wheel arches at the suspension. Springs can crack and break, most commonly at the top or bottom, so look there. The dampers should not be oily; look for 'wetness' in the area highlighted.
Quick Checks:
Dip stick, usually a ring-pull end that's easily accessible. Check the oil level while cold to make sure it gives the correct reading (if the engine had been running, the oil won’t have fully drained to the bottom of the engine therefore giving a lower reading).
Pull the dipstick all the way out, wiping it clean with a tissue, then re-insert it fully and remove again (not cleaning it this time) to get a fresh reading. There will be markings on the end showing a minimum and maximum level, the oil level should be in the middle. Check it's nice and clean by holding it up to the light and looking for any metallic traces in it. It should be a nice golden colour although some engines blacken the oil more than others. The oil in an older diesel car will go black very quickly for example.
Coolant cap, this can either be on the radiator or on a separate bottle (expansion tank) near it. Normally marked up with a caution as you should not try opening when hot.
Check the coolant levels too while the engine is cold; The coolant should be between the levels indicated on either the side of the radiator or the expansion tank. It should also be a nice clean colour, red, blue or green are common colours for coolant. If it looks brown or dirty then it may not have been changed in a while. Dip your finger in the coolant and check there is no oily residue on the surface giving the water a ‘rainbow’ affect.
Oil filter cap, normally marked up as 'OIL' or with a picture of an oil can and will be on the main body of the engine.
When you take this off, you're looking to make sure it's nice and clean, if you can see some engine components through the hole, they should also be clean with a slight golden sheen from the oil. Any black sludge type residue here can be a sign that the car hasn't been serviced regularly. Also look out for white creamy substance, it generally shouldn't be there, more on that below.
Engine bay basics: Although engine bays can look very different, there share some common features to make it easy to find the important parts.
In the images below, the red arrows point to either an oil filler cap (where you fill the engine with oil) and / or the oil dipstick (where you check the level of the oil).
The blue arrows point towards the coolant cap, either on the radiator or on a separate bottle near it.
Check the tyres for tread depth using the markers and general condition (cracking or 'old' looking) A good sign is 4x matching tyres, all the same make. It's not uncommon for the front x2 and the rear x2 to be different brands as they wear at different rates. However, 4x different tyres can result in strange handling characteristics and shows the owner has perhaps maintained the car on a budget.
The classic, Mayo! A quick google will show you hundreds of images of an oil cap with a mayonnaise type substance inside.
This is where oil and water are mixing inside the engine, which means the headgasket has failed.
However, a very small amount of 'mayo' can be a result of condensation mixing with the oil. If the car isn't used often and only used for short journeys where it cannot warm up fully, the condensation does not evaporate fully. An extended test drive to get the car properly warmed up is needed in these cases, maybe even with some 'enthusiastic' driving mixed in. If you return from the test drive and the mayo is worse, that's a problem. If it's gone, that confirms it was only condensation.
Detail Checks:
These are all things checked during an MOT but they're easy enough to look for yourself.
Brake Fluid reservoir, normally at the back of the engine. A quick check to make sure the level is between the marking should suffice here, no need to remove the cap.