Are there core characteristics one should look for?
Quite simply, we learn more from teachers we like. We like people who are like ourselves, or rather, like the people we wish we were. Simply put, if there's something that we find particularly lovely about another person, we're more inclined to do as they ask and find interesting the things that they say.
This works well for martial arts too since we're asking someone to both invade our personal space and that they respect the trust we're placing in them. There is, of course, a level of intimacy that goes along with this, which is why most people who genuinely like the school they're in spend a great deal of time there, or with their instructor or classmates in social situations.
Secondly, your interest in what he's teaching. If you have no interest in the subject matter, no amount of interest in the sensei will change that. This can be as grave as you simply don't find the art interesting, or as simple as his approach being sport while your interest lies in personal defense.
Thirdly, a knowledge of the material. It's rather a common con to see an instructor with lots of framed certificates teaching something that's passably some art, or would be to the untrained observer. If you take the trial lesson or sit in, observe, and attempt to remember the format that a technique took, then compare that with known high-quality instructors of whom video evidence exists (Youtube can be your friend!), you can determine at least reasonably if they appear to know what they're doing.
What kinds of questions should I ask to find someone that is right for me?
The problem with questions is two fold: if you do not know the answer, you simply need to know if they're lying. If you can not do either, then the question serves no purpose.
The basic questions that you should know are:
- How much does it cost? (This should be relatively in line with what others are charging)
- Are there testing fees? (This is a hidden cost)
- Are there contracts? (Again, these often contain hidden fees, including a cost to break)
- Do you offer a free trial? (Most, worth their salt, will.)
Anything else can be helpful, but these should be your priority.