Well, let me start by saying that I have been a martial arts practitioner for something like 15 years... I love it. Also, I've taken a whole bunch of different styles, both internal and external, ranging from styles which have focused on competition/ring sucess to styles which have been more of the "no nonsence/street self defence" varaity. So, that being said...
What would you expect of a good class/school?
Thats a tough one, as it depends entirely on what you, personally, are looking for. If I were you I'd start by trying to really acurately define what it is you are looking for in a martial art. Is it for fitness? Pure street style self defence? Just a new hobby? Want to focus on weapons or bare handed techniques?
After that, figure out if you prefer a "hard" or "soft" style. "Hard" styles are your more typical striking based martial arts. Tae Kwon Do, Boxing, Muy Thai, etc, all fall into this category, as well as Judo and BJJ. "Soft" styles are more along the line of Tai Chi or Aikido and focus more on the transfer of energy durring an engagement.
The expectations are going to be diffferent for each combination of goal and style.
What would you expect of a good instructor?
This one is easier, I think. A good martial arts instructor will not believe in the infalability of his particular style. He (or she, as is very often the case in a good instructor) will understand that it is not the style that makes someone a good fighter, but rather that it is the fighter who brings out the best that the art has to offer. They will also be creative, patient, understanding, diciplined and have very good communications skills.
Think of the best teachers you had in school. What charcteristics did they have in common? The subject matter is irrelevent; a good teacher is a good teacher.
What would your expectations be about the training and what things would you like to learn?
My expectations for training are very high. Because of my experience I dont want to go to a school that forces me to "learn" bascis that I have mastered several times over. If the instructor(s) is unwilling to let me quickly move on to more advanced techniques, I'll walk. I need to feel like I am learning, not just paying for the privledge of being there. Also, the vast majority of the time you spend in the class should be on actually practicing techniques/sparring. If you are spending 30 minutes of a 1 hour 30 minute class working on conditioning they are wasting your time. You are there to learn what only they can teach you. If you want to get better cardio, you can go running on your own time.
Personally I think that my ground technique needs work, so I would be looking for a good BJJ gym so that I could focus on take-downs/ take-down defence as well as submissions. However, I am also very much into the spiritual side of martial arts, and Aikido (if I can find a good school) focuses very heavely on that.
Edit: Spelling and gramar.