My first piece of gear was a dive computer... just something simple like Mares Puck or Cressi Leonardo. This way, even if you have to rent everything else, you know how your computer works wherever you go.
Then, I bought my wetsuit. Once I figured out where most of my diving would be, I got a 7mm wetsuit since most of my diving is Jersey shore or Dutch Springs.
Then, after trying some other gear, I settled on my BCD and regulators. Take this time to rent different configurations, both back inflate and jacket style (or if you have a mentor that's familiar with backplate/wing setups, try those too). Try to determine if Air2 or a standard octo would work better for you and your dive needs. Since these are your biggest investment, don't rush into buying them.
Once I had the basics down, I bought my own tanks. These were purely for convenience, so I don't have to drive across the state to return them after work on a Monday. These might provide value to you or they might not.
And of course, once I had everything else, I bought my own weights.
This was a logical progression that I took, it might work for you too. But my recommendation above all is to buy a computer (get one that's Nitrox capable, you'll be glad you did) and learn its functions inside and out. If you have to rent all your gear, regs all work the same, BCD's are all pretty similar, but interfaces on computers vary greatly. Buy the thing with the highest learning curve first.