First, you want to read the AKC Field Trial rulebook - which you can download on akc.org. My basic thoughts on the two below: (FYI, we're leaving for nationals tomorrow early am, so if I don't answer beyond this, my apologies)
AKC puppy - the judges are looking for a pup that is bold and unafraid, and knows why it is out there. The judges will want to see the pup checking logical objectives (treelines, bushes, deadfall, etc that is likely to hold birds) and using the wind intelligently. Playing with your bracemate is somewhat tolerated at this age, particularly at the cast off, but not for very long. It doesn't look good if the pup is more interested in playing than hunting. Tagging or aggressive behavior is never tolerated. The pup should handle fairly well - meaning listen when you give it a command. It should hunt to the front - meaning between 9 and 3 on the clock, the more forward the better. Hunting behind is frowned upon. The pup needs to range away from you, not hunt right at your feet. Each judge has a different opinion of what the appropriate range is, and your pup is supposed to be judged based on its future potential as either a gundog (medium range) or all age (HUGE range). But in general, if your pup won't range at LEAST 50-100 yds out, preferably 200 yds now and then, or more, you won't most of the time place. If your pup feels the need to return all the way to you many times during the brace, the judges are likely to question its independence. You also need to be able to see the pup and gather him/her up at the end of the brace. If you do encounter birds on the course, you shoot your blank gun after either you or the pup flushes the bird. Most of the time, you will not find birds in puppy as it is not typically planted.
AKC derby - all of the above holds true, although the judges will be looking for more independence and more forward run at derby stage. Your pup also must find and point at least one bird and hold it long enough for two things to happen: you must "call" the point - meaning, declare it to the judge, and you should be able to get within reasonable shotgun range before the bird goes up. If the puppy bumps, chases, grabs, retrieves in derby this should not be penalized after establishing point. Whether the pup bumps the bird, or you flush it (which is the goal) you shoot your blank gun. Of course, the pup should not show any concern about the shot. That said, if they consistently bump and chase without pointing, that will not be looked upon kindly. Again, the judges are judging ability and future potential. The derby dog is not expected to honor if it encounters its bracemate on point. The derby dog will be judged on its point - judges will look for intensity on point and will often give preference to dogs who have a high point with a high tail, be that right or wrong.
Hope that helps.