The most essential start-up 'equipment' you need is experience. What sort of grooming experience do you have? You will be responsible for the care and safety of someone's beloved pet. You will be working alone, so you won't have anyone else's guidance to help you. I may be stepping out on a limb here, but by reading your post, with you wondering what equipment you need to get started, it sounds like you have very little experience. If that's the case, then you should start by working in a shop or go to a reputable grooming school at the very least.
Old groomers never die, they just go at a slower clip.