There are lots of things to consider before breeding any horse so it pays to do your research :)
The conformation photo isn't the best, since she's looking towards the camera, isn't completely square in front and is wearing boots. However, she seems to have a decent shoulder and fairly well set on neck (again this is difficult to say for certain due to her position), but it does tie in quite deep. Can't tell anything about her gullet, front left foot may be ever so slightly clubby.. Pasterns look to be on the longer side and maybe a fraction upright. Would maybe like to see a slightly longer winter, but TBs are notorious for having tall and short withers lol Her back appears short but she is a fraction long through the loin and appears to have damage lumbar region. Since you say she was raced, it is possible that racing was the cause- I believe it if often caused from jumping out of the barriers but something to keep in mind regardless. Her main weakness in my opinion is her hind end. Her SI is located rather far back behind her hip, giving her a short croup and contributing to the long loin. Stifles and hocks are high compared to her front and she is also a little straight through her hocks. Ideally, you'd like to see a bigger hindquarter.
Movement wise, I think you'd really want to improve her canter. The stride appears to be on the shorter side, lacking cadence and she is a little 'pokey'- she doesn't really come through with her hind legs. This is most obvious in the trot. I would like to see a little more activity in her hind legs and a little more reach under her body, but I do like the freedom she has in her shoulder. Again, I think this comes down to her front being better built compared to her hindquarters.
I am not super familiar with American TB lines as I'm an Aussie, but I did notice that she has a pretty good dam line through her dam's sire. I don't know much of anything about her sire, but I do like the fact he has Mr. Prospector on his dam's side. Downside is that he is fairly far back in the pedigree when considering your mare and any resulting foal. Will have to look into these when I've got a bit more time. TB pedigrees can lead me down a rabbit hole which can last for hours lol
I believe if I am remembering correctly, your mare has had some soundness issues in the past? If so, these are so important to consider, since conformation can contribute to these. Can you elaborate a bit more here? I am a little suspicious of those front legs, mainly the lower leg due to her pasterns and the potentially clubby foot.
Now that I've picked your mare apart (and don't get me wrong, she's not a badly built horse. You've just got to be critical when talking about things like breeding to try and get the best possible outcome), what exactly are your breeding goals? You mentioned in your original post that your mare does hunters and lower level dressage (I'm not fully up to speed with the Aussie equivalent of the US levels, but 1st level is literally the first officially recognized competitive level isn't it?), which are pretty different from each other. You want a hunter to travel long and low and open with long smooth strides, whereas a dressage horse is expected to have a lot more sit/more compact and carry themselves more up in front (in a very very basic sense). A downhill hunter horse is not so much of an issue (in fact a lot of the top hunters I have seen tend to be on the downhill side where as downhill in a dressage horse can pose some training difficulties in the higher levels. So, what are you exactly hoping to produce with your mare? A top level dressage horse? A top level hunter? Or a general all-rounder? What level are you hoping to achieve?
All of this will or at least should impact your choices when choosing a stallion. You won't find many hunter stallions who have successful offspring in the higher dressage levels and vise versa. She is more a hunter type than a dressage type, so I would lean towards heading in that direction when selecting a stallion. If you can give some more info on this and anything else you'd like to 'improve' on your mare (like height or bone ect.) then it should help narrow down some stallion suggestions :)
This next bit is going to bit of a breeding info dump so just bare with me :)
Just deciding you would like a foal out of your mare unfortunately doesn't mean its a simple wham bam thank you ma'am and your mare is in-foal. Generally, any complications that can come up will come up; or at least plan for that to be the case and be grateful if they don't!
How old is your mare? Has she foaled before or is she a maiden? Older maiden mares are notoriously tricky to get in-foal and can require pretty intensive veterinary attention to make it work. Next after them are the older barren mares (those who haven't foaled in some time). This can limit your options for stallions, as if she is either of the above you will probably want to steer clear of frozen semen. Commonly, these mares have trouble clearing fluid from the uterus post breeding and coupling that with the lower mortality of frozen semen is usually not a good mix; often these mares require oxytocin shots after breeding to help clear that fluid before the embryo descends into the uterus. If your mare is fairly young, you could attempt frozen, but there are lower success rates and it is often more expensive than fresh/chilled because they require more intensive monitoring pre and post insemination.
Next the mare will require at least three scans to watch how the pregnancy progresses in the early stages as well as identify any twins/triplets/quads which may crop up and need to be dealt with. All things which can increase your vet bill!
Once your mare is in-foal, will you be okay with not being able to ride her for a series of months? Many stallion owners will actually void a LFG if the mare is brought back into work/competition after breeding, so make sure you read any and all contracts carefully. Actual care of the mare doesn't change too drastically until the last trimester, where her dietary needs increase greatly and increase again during lactation. If you're barn/stable doesn't have excellent pasture, expect a decent increase in your feed bill. Also, if she is kept at a busy barn where horses are coming and going, there is a distinct risk of disease being brought onto the property, particularly if they are attending large competitions. Some of these diseases can cause abortion in the pregnant mare so you will need to find out if you are in an at risk area and what you will do to minimize the risk such as vaccinations or isolation from traveling horses.
Come foaling time, do you have an area where she can foal down safely? Are you or a mentor/barn owner/ect. knowledgeable enough to recognize common problems before, during or after foaling? (ie. Placentitis, dystocia, dummy syndrome). Are there vets or equine clinic located nearby if something does go wrong? What's your contingency plan if something does go drastically wrong?
Majority of pregnancies/foalings go down without a hitch, but things can turn sour quickly especially during foaling. There is a very real risk of loosing the foal, the mare or both. If you lost your mare, would you be able to raise an orphan foal? That has to be a risk you are willing to acknowledge and accept. We had one mare at stud this year who had foaled down with no trouble previously have a horrific dystocia and then retained the placenta. The foal was dead and they nearly lost the mare too. A colt foal at the same stud presented with dropped fetlocks on his hind legs and has required a lot of care to help strength those ligaments and as racked up some pretty decent vet and farrier bills in his first month of life, not including the extra work required since he was required to be on stall rest and limited exercise.
I'm sure there are more things I have forgotten to mention (there always are in a topic as expansive as this) but realistically it will probably cost you a minimum of $5000 to get a foal on the ground, provided there are no complications, the stud fee is mid range (1000-1800) and everything goes smoothly. If you are not willing to risk your mare, do not have access to the required facilities or do not have the knowledge/are unable to find a mentor then it may be better to find a foal/weanling/yearling who is already on the ground and suits your requirements. Breeding can be expensive and heartbreaking, but also incredibly rewarding. Its just important to consider all possibilities before committing :) Research and talking to the people in the know is so important and when you think you know enough, read some more. You can never know enough when it comes to stuff like this :)