Firstly you need to decide what sort of animal you want and this probably relates to who will be riding it and what level of competitiveness, activities etc you are aiming at. Make sure it fits these criteria.
What you need and what DD wants will probably be very different. Are you all going to share it (DH included?).
Ask them to tack it up in front of you and then try it out in the menage and also hack it. If you plan to jump then have a go at jumping. If it does cross country see if you can find somewhere to jump something suitable.
I agree that you should take an experienced person with you - perhaps your trainer or livery yard owner.
Find out if it has any vices and make sure they let you know exactly how it reacts in different situations such as vet, farrier, loading, catching, mounting etc.
Has it done any competing - if so find out how it has done.
I would definitely recommend a vetting (by somebody independent of the vendors vet). Unless you are paying a lot of money a 5 stage is probably over the top. However a 3 part is worthwhile - particularly if you are inexperienced. If the vets are good then they will probably suggest taking a blood sample and this is a good idea. That way if you have any issues they will be able to check to see if the animal had been sedated or given pain killers.
I would suggest that,even if you are keen, that you walk away and discuss it privately with your family (over lunch?). You need to find out what you all think without being pressured by the owners.
Finally if you decide to buy then haggle. It is a buyers market at the moment and particularly so if they don't want to keep it over the Winter. You should be able to drive a good deal.
I would suggest that you may need to get your eye in as well and don't buy the first horse you try immediately although you could go back to it after seeing one or two more.