Here are some useful tips on many of the pitfalls - remember horses have been around a lot longer than cars, we all know what tricks there are to fixing cars! There are as many tricks and more to passing on a horse that is not as described!
To make sure you are ready for your own horse ride as many different horses as you can. Riding the same horse every time is not going to teach you anything about quirks of different horses.
It is no good asking "What breed?" there are as many differences between each animal of the same breed as there are in breeds themselves. Go for temperament rather than breed, colour, registration or anything else.
Just because you are over 5'5" does not mean you need a tall horse. Mark Todd at 6'2" was very successful internationally riding Charisma who stood 15.3. William Fox Pitt is 6'6" and rides 16.2 horses equally as successfully.
When you do start looking for a horse be honest with yourself as to your ability. Just because you once jumped 3'6" on a schoolmaster does not mean you are ready to ride a green young horse that is jumping the same height.
Always go to see the horse. See it in the stable, in the field, being ridden and ride it yourself. Take someone along with you that knows your riding ability and is very experienced with horses. If you can see the horse more than once. Ride it in the arena and out on trails. See if it is barn sour by riding it away from other horses and away from the barn. Have the vendor prove it is traffic proof, handle the horse in and out of the stable.
Ask what it is like with the farrier and to clip, load and ask about any health issues.
Some sellers will allow a horse to go on trial. Personally unless I know the buyer well, I will not allow this, any trial on or around the local area is fine but I will not risk a horse going off with someone I do not know.
Always have a horse vetted. Try to be present when this happens. I have known horses to have mild sedation when tried, then they vet goes along and the horse is dope free but ridden by an experienced rider and passes all health tests.
Ask the vet to draw blood when he examines the animal. One phial is given to the vendor and the other the vet takes. Both are labeled and initialled by vet and vendor. If the horse goes lame or is of very different characteristics to when you tried it, or goes lame, you can have the blood tested for either dope or pain killers.
If you are keeping your new horse at a livery barn then make sure that the staff are willing to help you. Continue to have regular lessons there is to much to learn to think you can manage on your own!
If you are keeping a horse on your own then again continue with the lessons. Make sure that you know a lot about the care and of maintaining manners on that horse. You can only do this if you have been use to handling a lot of horses.
Never be afraid to ask for help.
Owning a horse involves a lot of commitment as well as expense but the rewards can outweigh everything else.
I am sure many others have more to add.