Welcome to Scuba and the boards. I know a lot of people who bought gear during the class. I have mixed feelings about it. Other than personal gear (mask, fins, snorkel, wetsuit which I had as a snorkeler before certification) I waited until after a couple of dive trips, that way I was able to try some things out. I love having my own gear and did have minor issues with rental gear but nothing more than just a nuisance. I am not sure what your financial situation or geographical situation is either. If you are about to go diving or live somewhere where you can dive frequently, then I would say buying gear early is a good idea from a fiscal perspective. If you won't be diving much you can also see how the sales go - you just missed a couple of great scubapro and apeks ones (maybe there is some stock left floating around). There are usually killer Black Friday HOG/EDGE sales among others. Here is also the order I purchased these items.
Computer, I think for the dive computer if you want one, something simple and affordable is a good solid purchase. Then if you want something really fancy later this can become your backup computer. Think Suunto Zoop or similar ( I got a new Cressi Leonardo from LP on black Friday for $175)
For a regulator - you need to decide what your needs are vs. your budget (cold water, # of ports, adjustable?). I have purchased like new used regulator sets and had them serviced for a really great deal. If I hadn't done this I would have jumped on the apeks or scubapro sale or gone EDGE. It sounds like there are a lot of common favorites floating around. HOG/EDGE, Scubapro, Apeks/Aqualung, Zeagle and Atomics always seem to be pretty highly recommended. What does your LDS sell and service? Without knowing more about you it is hard to recommend specific models.
BCD - Fit is really important, and also what types of diving you are doing, ie I don't know your specifics so its hard to say. I did get a BPW as a new diver and had dove rental jackets and back inflates and occassionally get in one at the LDS. I love love love my BPW and am a new rec diver, so while I can recommend this to you, I am not sure what your needs are beyond that. The nice thing about BPW is it is modular and can be customized and then you can switch things in and out if your needs change later. I did get mine from DSS and Tobin was great at helping a newbie figure out what was needed, how to set it up, great supporting info, too. When my hubby is ready to buy he will be calling Tobin. That being said, there are lots of choices in BPW. Dive Right in Scuba, Northeast Scuba Supply, and Jim Lapenta also come to mind as great sources.
Once again, welcome to this addictive and wonderful sport!