Hi Ken,
Welcome to ScubaBoard, and congrats on your certification.
Here's the order in which I typically recommend new divers buy their gear:
1. Personal gear (mask, snorkel, fins, booties)
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2. Exposure suit
3. Computer
4. BC
5. Regulator
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6. Tanks, SMBs, slates, lights, reels, etc.
You normally would have had to buy personal gear (1) before even taking your class -- but I suppose you haven't. Those pieces have to be quite precisely the right sizes for your body to be comfortable and reliable. I put personal gear in its own category, because it's a "must buy."
The wetsuit (2) is next, for a couple of reasons. First, rental wetsuits are almost universally nasty. They're bleached out from chlorine, are full of holes, and are always the wrong size. A wrongly-sized wetsuit is hard to doff and don, and will not keep you warm. You'll be amazed at how nice it is to own a wetsuit that fits you perfectly. In addition, depending on your normal water temperature, invest in a high-quality hood and good gloves. I recommend a two-piece 5-7mm suit, 5mm gloves, and 7mm hood.
I put computer (3) next, because many recreational divers love them, but most rental computers suck. You're better off getting a little computer you like. I recommend a hockey-puck computer, wrist- or console-mounted. I do not recommend air-integrated or hoseless air-integrated computers.
You might want to wait on the computer though, and just buy a cheap bottom timer/depth gauge. Tables still work.
Next, get yourself a good BC (4). Opinions on BCs are as widely varied as opinions about cars -- you need to find one that fits you and your diving needs. Try a bunch out in a pool, if you can. I have to recommend a backplate and wings for comfort, utility, and simplicity, but I understand that BP/wings are not attractive to many divers.
Regulators are near the end of your gearing-up process (5), because they're expensive, and most respectable rental agencies rent pretty good regs these days. Unless you dive often, you might not need to get your own for a while. Just make sure the regs you rent are well-maintained -- they are life-support equipment, and you shouldn't "trust" anyone. If you start doing anything other than open-water recreational diving, owning your own regs is an absolute must. In addition, you might really enjoy the easier breathing and higher comfort of your own high-performance reg. Some divers take some acclimation before they're capable of discerning the sometimes subtle differences between low-quality and high-quality regs.
Next, there's everything else (6). I've put it all in a category by itself, because some of these extras are very cheap, and you'll want some pieces before others, based on how you dive. Surface marker buoys and signaling devices are important for ocean diving, so you should get those before jumping on a boat. Lights are obviously important for night diving. I wouldn't advise that you buy your own tanks unless you get into mixing, or there is no dive shop near your intended dive spots. Most of the time, tanks are so cheap to rent that the expense to purchase and transport them is unjustified.
Hope this helps!
- Warren