Even as a new diver, you want to be a fully equipped diver. You will be more comfortable and safer in your own gear, and you ultimately dive more.
Most certification courses require you to have at a minimum, your own mask, snorkel, fins (and boots, depending on your type of fins). Most also consider a wetsuit part of your personal gear that you want to ensure fits well and offers adequate thermal protection for the environment you will typically be diving in. For ocean diving, grab a surface marker buoy as well.
Following these basic pieces of gear, I would recommend a computer. Most divers still learn dive tables, so it is possible to dive with only a watch suitable for depth and a dive table, but the square profile the tables assume, significantly limits your available dive time. Especially if you plan to travel on dive vacations, with multiple dives over multiple days, you will gain significantly more bottom time with a computer versus a watch and RDP. If you aren't buying your life support system (regulators and gauges), there are several great wrist-mounted computer options.
Next I would recommend your life support system. For your second stage, you can buy configurations with venturi control and/or spring tension adjustments that allow you to fine tune your breathing effort. You generally won't find these features on rental gear. When buying your regulators and computer at the same time, you have options for air integrated computers on your console or wrist.
The final piece of the kit is the BCD. The biggest benefit to owning your own BCD versus renting is weight integration. Rental gear it typically a jacket style BCD and a weight belt. Owning your own gear allows you to chose different styles of BCD from jacket, to back inflate, or a hybrid of the two. Back inflate BCDs help with horizontal trim in the water and so does positioning the weight in integrated pockets and trim pockets versus a weight belt. Not to mention, you will be more comfortable in a BCD that fits well and you will know where all the dump valves are and how to operate them. If you drive a car, you won't have many problems in a rental car. But you are still more comfortable in your own car...no turning on the windshield wipers when you meant to turn on the headlights.
Once you have your full kit, you still have plenty of choices for accessories like lights, dive floats, reels, cameras, dpv, ...