Ah! A great question! Typically in looking for any horse, a lease horse, a horse to buy, a stud, etc. you should look for these things:
- UTD on shots, vaccines, etc?
- How experienced is the horse?
- How old, last time it has been ridden, training needed?
- Allergies, health issues, past health issues, issues with the family line?
- Rescue, any odd vices?
- Have a vet check up on it.
These are just some things. I recommend googling it (which, you probably already have). And make sure to try the horse out, etc. Take your time in deciding, it will take a long time and will not be the first horse you see. Try looking on FaceBook if you have it, etc.
Clear x-rays and generally good health would be the beginning.
It really all comes down to what discipline is the horse for. I'd recommend you find an experienced person to guide you through this. As for the horse's anatomy I've found it useful to look at horses walk and canter because those are the gaits you can't really change, you can improve them tho. Where as trot is completely "moldable" you can teach it however you want. It will take time, but it's doable. Don't forget to check all the proper vaccines and teeth condition as well as the horse's age. It's important that your first horse has a good character and isn't aggressive. And another thing would be that special feeling, when you know it's the one! Good luck!