This question can be answered very differently depending on your goals. Is your goal to add one piece of equipment at a time, without having to worry about keeping anything alive? Looking to keep things alive and thriving, or just alive? For my answer, I will assume that you want to provide the bare necessities of life.
First off, you can't keep fish or corals alive without oxygen, and corals have the additional requirement of light. So at the top of my list I would put water circulation (be it a return pump that circulates water to the display from the sump, or powerheads in the display) and then lights. After that - a heater to keep the tank from freezing and then a skimmer to clean the water.
I would rank ATO and controller in last place. I ran my tank for a long time by just topping it up with a jug of water every day, and I still run without a controller.
Those will be the basics for keeping a saltwater tank. Depending on the difficulty of the livestock you want to keep, the better/bigger you will want to go with your equipment choices. For instance if you want to keep SPS then go as powerful as you can for the lights, as strong as you can for the water circulation, and as big as you can for the skimmer. If you want to keep soft corals, then you can go with bottom of the line equipment.
