The only thing you really "need" to own is a good mask - one that fits your face well, is comfortable, and doesn't leak too much. You can certainly order a mask online, but I wouldn't. You need to TRY THEM ON to make sure you like the fit...and that's a little difficult over the computer.
Oh, buy a snorkel when you buy your mask. Mine usually stays in my gear bag, but if I'm doing a rockin' drift dive where I could get separated from the boat or a shore dive that requires a surface swim, I'll break it out. They make collapsible snorkels that fit in your BC pocket, but I've never used one...think I'll try one out sometime, because that sounds like a good idea.
Anyway...
Once you have your mask, everything else CAN be rented...
...but if you want to keep buying, fins/boots would be the next logical step.
I'd recommend waiting on those until you figure out what you like - full-foot fins or open-heel fins with boots. If you NEVER plan to dive in cold water, you may prefer a full-foot style fin, as you don't need boots and there are fewer items to keep track of and, ultimately, lose. If you want to do any diving in lakes/rivers in upstate NY, you'll need to wear boots or your feet will freeze off, so that means open heel fins. Try as many styles as you can before you decide what you like - this means getting in the pool with different styles and seeing which ones are most comfortable to kick.
Once you have a mask and fins, the next thing I would consider is a regulator/computer set-up. This will be the single most expensive piece of scuba equipment you will ever buy, but if you maintain it properly, you should only have to buy one every 15-20 years or so. You'll get lots of opinions on brands/styles, but IMHO the most important thing is to make sure that what you get can be serviced locally. You need to have yearly factory-sponsored preventative maintenance done on your regulator in order to maintain the warranty and that's a complete pain in the @$$ if you have to send it away to have it done.
The last major thing I'd buy would be a BC. This is all about fit and what you like best...except for one small thing - which may not make sense to you right now - but bear with me for a second here: Whatever you do, DO NOT get your alternate air source integrated into your BC power inflator. No matter who tells you it's a good idea - no matter how much your instructor/dive shop owner tells you it's a good idea - no matter who tells you how great it is to eliminate a hose - DO NOT DO THIS. It is a universally terrible idea that the scuba community needs to forget IMMEDIATELY. If you want my reasons why, PM me and I'd be happy to tell you.
Now, you've got everything you really NEED...and are a few thousand dollars lighter in the wallet.
Hey, you asked!!!
