What attributes are most important - this can depend on what discipline you want the horse for but generally the two main attributes we want first are: soundness and attitude/temperament.

What next - don't be afraid to ask to see the horses feet picked up by the owner - the horse caught in a large paddock - the horse put on a float and tied up.  This will tell you if the horse really is 'good to tie up, float, shoe and catch.'

When the horse is being saddled watch to see if the horse gets agitated when the girth is done up, or if the horse is difficult to bridle.

Always ask the owner to ride the horse before you ride the horse.

Come back a second time before deciding to purchase a horse. On a second look you will notice things that you didn't see the first time.

Rub the horse all over its body to see if it has any spots it doesn't like being touched - e.g. ears, belly, tail, legs etc.

Ask to see the horse lead to ensure it doesn't push or drag the handler. 

Vet checks: whether you get a vet check can depend on the discipline you are interested in or the price you are paying.  While some horses won't pass a vet check this doesn't mean you shouldn't buy the horse but will indicate any problems the horse may have.  This can also help to verify the age and general health of the horse.

Make a list: make a list of all the attributes you see as mandatory and those that are nice to have - use this when gathering information about prospective horses to look at. Try and stick to horses that only meet your mandatory requirements.

Don't buy horses that are in poor condition, sick, unsound or have a bad attitude unless you are absolutely certain you know what you are doing.