Since you're wanting more information I'm going to post an answer to try and help you move forward with your training and your goals. I'm going to have to assume you have no prior training in any martial art and would be considered a novice in your training.
This answer will 100% reference Dave Liepmann's excellent answer.
The first and most important part is don't try to do it on your own. There is a reason all martial arts developed in groups. As a developing student you will learn many of the rights and wrongs from a skilled and informed trainer. Without one you will struggle, develop bad habits, and most likely become disheartened. You have an opportunity to learn from someone and I recommend giving it a shot. If the trainer you are working with does not seem to provide any valuable training effort or seems like he/she does not have much confidence I would not move forward with their services.
The other part of Liepmann's answer is there are other routes. Train! Train any skill that gets you in shape. Focus not on punching, kicking, or martial arts in general. Just get in shape and be confident in yourself. Continue to look for a trainer until you find your right fit, but until then develop other skills that you will inevitably need in your advancement.
Lastly, I sadly do not have any online resources. Though they could be there and may work for some I just never thought they were worth it. I had an in house "dojo" experience. I spent most my life being told my mistakes and then passing on my learnings to students. The most important part of that experience is having some one right there constructively telling me to do X when I thought I was doing X, but in fact doing Y. When you get a chance to experience that you will not want to watch a DVD or YouTube video.
I will give one thing for a proper punch that I feel is the most important.
Take your arm and lift in front of you. Point your index finger to the center of the monitor. At this point your arm should be straight and your finger pointed directly to center of monitor. Now close your index finger so that you make a fist. (Touch base... Your fist should be closed with your thumb OVER your index and middle fingers) Continue to imagine you're pointing your finger at the monitor.
If you notice in doing so your now closed index and middle fingers are the forward most knuckles. A proper punch will directly hit an object with those two knuckles almost exclusively. If you now try to point ANY other direction other than at your target (center of the monitor) you will quickly see what an incorrect punch looks like.
The reason you punch like this is because the way your hand is designed you have the full support by the bones in your arm to structure around protecting those knuckles from impact. Sadly the rest of your hand does not have that structure and will likely crumble even on a minimal hit. Save yourself the pain and time and practice pointing correctly and when you get a trainer have them verify and build on what I have told you. Even if they describe it differently.