Stores like Petco & Petsmart have their own training programs and can get you started as both a trainer and a groomer. I do not have experience earning certificates via chain stores, but I do know they require you to sign a contract and non-compete in order to stay in their program.
You can also express interest in grooming by getting a job at a reputable grooming salon and simply asking! You will have to start off as a dog bather, but usually there will be some groomer that would love to show you the ins & outs.
I’ve heard okay reviews about Animal Behavior College. They have both dog training and dog grooming courses and include actual hands on training. There’s also Karen Pryor Academy for dog training, but in the end what you want is to be certified by the right organizations. Taking courses and gaining experience is key, but it should all lead up to being APDT & CPDT-KA Certified (in the US) for dog training.
I’m not sure if there is a certification process for dog grooming? Honestly certifications aren’t even required to be either, but it gives you the extra push into the career you want, while also learning important information. Learning by experience is great, but it is important to take proper courses to learn specifics. Especially when it comes to the well being of someone’s pet.