Hi Snorkel,
Most places require you to buy mask, fins and snorkel up front, and then they will rent you the rest of the stuff you need. However, they are in the business of selling gear and so expect them to encourage you to buy all your other gear too. Also, it's good to learn with the gear you will be diving with because with familiarity comes confidence.
In my case, I was scheduled to start Open Water training with my wife 16 years ago when she got pregnant (somehow!). Of course, we put diving on hold. At that time, I had bought new mask, fins, snorkel and booties, and was planning on renting the other gear for awhile. This time around, I splurged and bought all new gear for me, my wife, and two sons before we even hit the pool! It was pretty expensive but well worth it because I couldn't even imagine renting all that stuff, diving, then bringing it all back the next day. But in our case, my wife and I were finally fulfulling our dream of getting certified, AND... turning our kids on to diving at the same time. So it was a pretty easy decision.
Good luck, and let me know what you decide!