When choosing a horse, for a beginner, what are some characteristics or behaviors to look for? Are there certain breeds to look at or avoid and, if so, why?
So, we probably need a little more information as to your situation to provide a better answer. However, I can probably point you in the right direction.
First, what kind of beginner are we talking about? Is it a child or an adult? Do they have any experience with horses at all or is completely brand new to them. Have you figured out the logistics of the horse, including price up upkeep and whether or not it'll be at your house or if you're boarding. There are pros and cons to each.
I'll throw it out there and say that if the person getting the horse has absolutely no experience with horses, or even very little, then I don't suggest getting a horse at all. I suggest taking riding lessons and helping out at the barn for at least a year.
A horse is a very large animal. It isn't a dog where you can just put it in the back yard and if you don't teach it manners then just keep it locked up when company is around. If you don't know basic management practices when handling horses, then they WILL take advantage of you and escalate their bad behavior into something that can get your hurt. Even people that have been around horses a long time, don't often notice when a horse needs to be corrected. For instance, a horse will often step into your space. The horse needs to be driven out of your space. However, many people step away. In a horse herd, the one who can move the others feet is more dominant and is in charge. Simple things like this escalate, because the horse gains courage each time he doesn't something and nothing negative happens. This is the same principle as training. It has the potential to get your seriously hurt or even killed.
That's why I suggest taking lessons for at least a year. You'll be taught basic handling practices and much more important, you'll become familiar with horses. Over time, with association, people become familiar with situations to the point they can predict what's going to happen. It's not just with horses or other pets, but anything. You know that if you're driving on the road and someone comes up beside you, driving erratically, you slow down and let them pass. It's the same with horses, as you become familiar with them, you'll learn to see what they're doing and know what may happen next. This allows you to be prepared and have a much better chance of controlling the situation.
This time at the barn, helping out, also allows you the opportunity (or the child as the case may be) to determine whether or not you want commit to the level of work that is necessary. You need to scoop stall twice a day, go out and blanket horses in the field when it's starting to sleet. You need to get over the period of time where everything is still fresh and you're excited to go out and do the drudgery. Let it become a grind and see if you're still committed. This grind increases when you bring your horse home. Not only are you 100% responsible for all the chores, but you also have to maintain the grounds. This means fixing fence, mowing pastures, etc... Just those two at minimum require a lot of time, not to mention equipment. You need the fencing plyers, channel locks, staples, hammers, stretchers, etc... You'll have to purchase and maintain a tractor and bush hog.
That brings me to the next point, which is money. You need to figure up the costs of owning the horse over its expected lifetime. This will include farrier work, initial investment in tack, boarding, maintaining your home grounds, yearly shots, emergency vet visits, a truck and trailer to take your horse places, maintenance on all of those things, etc... Like anything else, there is give and take. If you board, it seems expensive, but you aren't buying a tractor and maintaining it, or repairing fence, or anything else. They can take 100% care of your horse and you can just pick them up and ride. You don't have to have the best equipment to maintain your fields, but you will need to know how to fix and maintain it if you get older equipment. So it's give and take. I suggest you set up a spreadsheet and figure up the actual cost. Factor in your time, which is really worth a lot. I probably spend a bergilliondy hours in care for every hour I get to enjoy my horse.
You should also figure in potential change in lifestyle. That's why I asked above, whether it was a child or adult. Some people can sell their horse if their situation changes, but many people get attached and make huge sacrifices when their life situation changes and it becomes hard to care for the horse. It's not a good market right now, either. I know people who have gone on trail rides and come back to discover two horses loaded on their trailer. No one wants a to buy and some people can't keep them.
So if you buy for a child or an adult who might not have a stable situation in a couple of years, that's something to consider. A child is interested now, but what about when they can start driving and hanging out with their friends. Even if they do, what about when they go to college. You still have to care for the horse and pay the bills.
So that's all my warnings and things to consider. If you've looked over all of that and still want a horse of your own, I'll cover that now.
The first thing to do is look at site after site where they advertise horse. Take down notes on any horse you like. What'll start to happen is you'll become familiar with the grade of horse you want and what prices they're going for. Just like on ebay or amazon, just because horses are marked at a certain price, doesn't mean they're selling at that price.
You asked about a particular breed and the answer is, no, it doesn't matter. I don't care what people tell you, there is a horse for you in every breed. Horses are individuals like people. If you want an Arabian, then get an Arabian. Sure there are plenty of hyper arabs, but there are just as many calm dead broke ones. A lot of it is stigma, like with pit bulls. They're perceived a certain way, so they're treated that way. People think arabs are hot and at many of the shows, they like to see that high energy and drive. So people think that's just the way they are and when they start acting fractious, they don't do anything about it. If the horses are trained to calm down when they experience something scary, then they'll be a calm horse. So you'll be able to find fiery eyed devils, dead broke kids horses, and everything in between. You just have to keep looking and not be willing to settle. The only caveat I'll add about not getting a certain breed is size. You can't put a large person on a tiny horse for the most part. Whatever size the horse is, you don't want to be more than 20% or so of that horses weight, including tack. Of course that varies like anything else and there are horses that can carry more or less weight, but that is a decent guideline.
Once you have sorted through adds on a few hundred horses, start considering what your goals are. Do you just want a horse to walk around the arena, go trail riding, do dressage, or anything else. Your goal will help you determine what to get for your first horse. To be honest, it really doesn't matter what your first horse is capable of. Most horse will be able to do low level of any discipline you want to do. Also, no matter what you think your goals are now, you'll probably change them some as you learn more about your horse and yourself. Once you start settling in to what you want to do, then you can get help deciding if the horses you're looking at have the confirmation and are at a good age to care you through that disciple till you're ready to advance.
You'll run into a lot of saying when it comes to horses and one of them is "Green on Green, make Black and Blue". What that means is that many people want to buy a young horses so that they can grow together. The same thing applies to rescue horse. People want to rescue a poor, abused, damaged horse. These are not the ones for you. It's the same reason why you don't let children teach other children in school. It's the reason you don't ask a drummer to teach you guitar. I'm not trying to say anything bad about you at all, but some one who is ignorant (not stupid) can't teach someone who is ignorant. Ignorance is just a lack of knowledge. You need a really solid horse who will take care of you until your knowledge of horses increases. Once you have enough knowledge, then you can bring a more inexperienced horse along, the same way your horse brought you along.
You don't need to narrow your selection list down to one horse, even if you're 110% sure that's the one you want, because you just can't tell from a picture. You should have several horses you want to look at. When you get to that point, you should find someone whom you believe is a knowledgeable horse person that is willing to help you. They will see things that you don't even notice or would know about if you did see them.
You should set up appointments to go see these horses. I'll warn you ahead of time of the biggest trap people fall into. That's love at first site. You've wanted a horse for so long and you're finally in the presence of one. It can be overwhelming. Don't let it sucker you into buying the first horse you come across, even if he turns out to be the one. That year of interaction I talked about earlier will go a long way toward alleviating this, but won't rid you of it entirely.
In the same vein, don't go for a horse just because it's pretty. Pretty can kill. The order of things you should look for in a horse are, temperament, confirmation, age, and looks.
Temperament is the horse's personality. Is he confident? Is he inquisitive? A great horse will still spook, it's whether he flips out and runs off bucking or stops to investigate the scary thing that will let you know if he's a good horse. You want a friendly horse that isn't too pushy (though that can change if you don't correct his attempts), that's friendly, inquisitive, and investigates what startles him.
Conformation is how the horse is build. His bone structure and his lines. You'll need an experience person to help you with this, and like I said, it's not super important for your first horse. Equis magazine used to run a good series where they'd give you three similar horses and you had to judge them based on confirmation. The next page showed you the order the judges put them in an why. I don't know if they still do that or not, but if you can find them, they're a great reference.
Age isn't really important, but it does matter some. I have a horse that was trustworthy in almost any situation when she was 2 yrs old. She was ridden in supper rocky mountains where she could run across wild ponies, bears, etc... She's a solid horse. However, most horses have been ridden enough and settled into their ways by the time they're 5-7yrs old. It's just like a dog or a child. They loose some of that wild rambunctious energy that can be annoying and can be trusted with a little more responsibility. On the other end of that scale, if you have a horse that is over 12yrs old and it still has bad habits, then it's probably not for you. It'll take more work, that you won't know how to do, than it's worth to turn them into a solid horse.
Lastly is looks. If you've got a few horses that have good temperament, descent conformation, are a good age where you can enjoy them for several years, then by all means, pick the prettiest one. Just let that be the last consideration. Trust me, there are enough horses out there, for sale or adoption, that you can meet all your criteria and still pick a "pretty" one. Just be patient. You're committing to what is possibly a 25-40yr relationship. Don't feel like you have to rush into it.
Take your experienced horse friend to check out the horses with you and when you think you have it narrowed down to one, get a trustworthy vet to check on the horse for general health. Tell them to look this non-gift horse in the mouth and make sure he's the age they claim. That's another saying you'll hear, "There is no such thing as a free horse". That's super true. The initial investment is nothing to the overall cost.
Sorry this reply was super long, but I felt it was all relevant information that you need to have if you're looking to purchase a horse for the first time. Horse friends will be your biggest resource, because horse people love to talk horses and they'll not only be able to help you know what you'd need to buy, but they can offer info on what good prices are, and help you choose your horse. It'll be easy to cultivate these friends at that year of riding lessons I was talking about. Also, you should feel encouraged to take lessons with your own horse after getting him. You'll still be in for a huge benefit. Good luck.
Edit: One last thing I thought of on my way back from lunch. If the person getting the horse has a handicap, be sure to consider that in your purchase. They need to be able to mount and dismount the horse. There are aids to assist with this, like mounting blocks. The horse can also be taught to park out. However, if the horse wants to misbehave and you're on a trail ride in the woods, they need to be able to mount. So the horses height would be a consideration there as well.