Unless your dog has a bleeding disorder (and you probably already know that), the nail will stop bleeding on its own. Any of these simple solutions will help stop nail bleeding. You can try them individually dry or make a paste out of them to make it easier to apply. Just add a few drops of water and you'll have a homemade astringent paste.
The task seems simple enough, especially with the wide range of nail clippers available today, but the procedure can go very wrong with a misplaced cut, leaving the dog scared and reluctant to allow you to get close to his feet again. It seems that your puppy was not carried away by the experience, but if he seems wary of cutting his nails in the future, be sure to combine holding the paw, holding the trimmer and then trimming his nails with positive things such as treats, toys or attention. In my experience, some dogs don't react at all when the fast one is cut off, and other dogs may walk away or give a little scream. Keep in mind that, while it is true that a Dremel cauterizes when it shortens the nails (so there is no blood), the dog will continue to feel pain if its nail is “too short”.
Nails that are cut too short rarely get infected, but there may be something else going on that started when the nail was cut. It might even be a good idea to have your nails cut for a few months, until the length of your nails is more manageable. I was so scared for my dog that I couldn't be home when the hairdresser arrived, but apparently it caused one of the nails to bleed a little, not for long. To prevent your dog from licking his foot once the bleeding has stopped, consider wrapping him up for a few hours, putting a sock over his foot (assuming your dog doesn't try to chew or swallow the sock), or distracting him with attention, toys or treats (as long as he doesn't get too excited).
You should always have astringent powder available and nearby to trim your nails in case your dog's nail starts to bleed. Hopefully, by cutting his nails repeatedly, your dog will begin to learn that he has nothing to fear. If your dog has transparent and other black nails, use an average transparent nail as a guide to cut the black ones. Fortunately, most of the time dogs don't try to lick their nails and rarely get infections from a toenail cut that's too short, but I understand that you want to be careful.
If you use nail clippers for dogs that crush your nails, have a safety guard (a “guide” that gives you a false sense of security), or don't give you enough control when trimming, I understand how frustrating it can be. However, as an integrative veterinarian with more than 20 years of experience (including teaching countless dog owners how to cut dogs' nails), I can assure you that you can avoid future setbacks by cutting your dog's nails and re-trimming your dog's nails with confidence.