I'm not a dog groomer, however I do know former dog groomers. At Petsmart grooming, they train you for it. Many of the people that become dog groomers were once hair dressers who moved onto grooming animals, so chances are the other people looking at the same job will be more qualified than you when it comes to that, although many grooming establishments do training. For dog grooming, other than the establishment itself, i'm really unsure about the training and whether you can go down the actual dog grooming avenue first, or study to become a hairdresser and move onto grooming.
The negatives and positives are pretty straight forward. Positives are that you get to work up close with the animals and gain the skills necessary to succeed and become an excellent groomer. Negatives in my opinion, outweigh the positives. It is a dirty and hard job. Expressing anal glands is one of the dirtiest aspects. Some dogs aren't well mannered when it comes to grooming so not only do you have to fight with the animal to keep them still so you can clip them, but you also have to do a good job in the process as they are paying for the services. Some dog groomers get more dogs to groom than others so the amount of pay can be steady or it could go up and down. If you're looking for a job within the animal field, dog grooming is definitely possible and with training you can succeed, but I personally think your time, patience, training time, etc. could be better spent elsewhere. Good luck!