Definately take someone who Knows horses (trainer etc.). Also line up a couple of horses that you like try them out ride them, spend time with them. If you can take the one you like best for a trial period that's great to. But the BIGGEST piece of advice I can give you is get a good vet lined up that is of Your choosing. Then once you find a horse you like have some type of Pre-Purchase Exam done. These exams can start out pretty basic from just a basic lameness exam to a full nose to tail exam with xrays. Depending on what you are looking to spend on the horse can help you determine what you may want to spend with the vet. Talk with your vet and see what they say. I would at least suggest a good exam of the horse with a lameness exam. Then if you decide this is the horse for you have the vet pull blood to store for a month. Your vet can see things that even a trainer may miss, they can also check the horse for old surgery scars, check the mouth and see what condition the horses teeth are in. Pulling the blood protects both the buyer and seller of the horse. THis way if you get the horse home and a couple of days later the horse goes lame or becomes crazy (so to speak) you can run the sample and test for drugs.

I have seen a lot of clients buy their first horse get it home and then have a vet look at it. Then the vet finds something wrong and it ends up costing the new owner $$$$ to fix the problem if it is fixable at all.

I would also try to stay away from horses traders.

I would say look for a more seasoned horse (older). For a first horse I would try to stay away from the younger ones. Look for something 8-10 years and up.
Horses live a long time with proper care. I lost a 30year old last year which I rode the weekend prior to him leaving us, and I still ride my 27year old 3 days a week out on the trails.
You want your first horse to be fun and to be able to enjoy it, So get a good vet and a knowledgable person lined up before you go out looking. There are a lot of horses out there to choose from so take your time and don't let the seller push you into things.

Good Luck and Happy Horse Hunting.