So much depends on your goals. Are you looking for a vegetable garden that you can eat and can from? Or do you want a flower or other decorative garden? What sort of space do you have to work with? How much time do you think you'll want to devote to it? What about your location--does it limit your growing season or what kinds of plants will grow there? Do you want to start from seed, or do you want established plants for your garden?
My first attempt (as an adult) at gardening was just a little too ambitious, and I didn't really do much planning. I grew everything from seed, rather than established plants, in a garden plot that was far too big for me to be able to keep weeded, in a section of the property that just wasn't suitable. As a result, the garden didn't fare well. It didn't have the right lighting, soil, or much else going for it, and it was too far from the house for me to really give it the attention it needed. I did get some produce from it, but not nearly as much as I could have had I just taken a little time in planning beforehand.
Regardless of what kind of garden you want to do, I'd suggest starting small just to be sure you actually want to garden, and that you have time to devote to it. In that case, you can do container gardens on a patio, or even in a good windowsill.
Once you decide what kind of garden you want, do a bit of reading. The thing about gardening is that you can make it as simple or as complex as you want, and most garden-type plants are fairly hardy and can survive a beginner's mistakes to a large extent (the two exceptions that come to mind are over watering and over fertilizing), but you'll be much happier if you learn a little about the plants you want to grow before you find out the hard way what they need to thrive.
One thing to remember is that while nurseries are usually good about not offering plants that are out of their growing zones, places like superstores aren't nearly so careful. You might end up with an exotic plant that survives in the store because it's sheltered, and find that in your garden it dies immediately. On the other hand, nurseries can be more expensive, but you usually get a lot of support from the staff. Just tell them you're a raw novice, and a good nursery will have people falling all over themselves to help you. They love sharing their passion.
As for supplies, that's going to depend entirely on your goals and space.