Who doesn't like saving money or getting a good deal? If it's going to cost $350 for basic gear to get started and $100 to rent the remaining gear whenever I want to dive ...or... $2,000 to own my own equipment....can I buy used equipment and save myself some money?
Let's assume you were able to find everything used, how much would you save? Probably 25-30%. Scuba gear is generally either worthless on the used market or holds it value very well and thus not much of a reason to buy used. If you planned to spend ~$2,000 you could probably save maybe $500 if you bought used. So now it becomes should you buy used equipment...
A mask is a personal thing and everyones face is a different shape. It is unlikely, dare I say impossible, to find a mask on the used market that properly fits your face unless you know exactly which mask you want because you have tried it on before.
Fins
Not much can go wrong with fins...if it's the kind you want. If the fin uses rubber/plastic heel straps, instead of metal, you will want to inspect them for miro cracks or damage. They do break on occasion, though most fins have replacement heel straps available to purchase after-market.
Snorkel
Absolutely...and then you'll probably never use it after your open water class. So buy a cheep one.
Wetsuit
Unless the wetsuit has less than a handful of dives and it comes from someone you trust...I wouldn't. Wetsuits wear out. They lose their buoyancy and warmth with time, and they do not have replaceable parts. Lastly, a proper fit is key. Some wetsuit manufacturers offer dozens of sizes, which means finding the perfect fit on the used market in good condition can be difficult unless you know your size in that brand.
Gloves
This would be less risky to buy used than a full wetsuit, just make sure the size is correct for your hand and the seams look to be holding.
Hood
Other than avoiding anything that looks to be old, you should be fine. However, I don't see hoods on the used market very often.
Boots
Other than avoiding anything that looks to be old, you should be fine. However, I have found SIGNIFICANT variety in sizing expectations so be careful.
Regulator Set
These hold their value very well so you may not get much of a discount for used, unless you go very old or very used. Make sure whatever you buy can be serviced by a technician at your local dive shop. And for heaven sake...have it serviced before you dive it.
Computer
Not much can go wrong on a computer so long as it is not "flooded". You can have it serviced at your local dive shop for new ORings and battery.
Flashlights
On the low end of the cost curve, there is not much of a used market. No harm in buying used if you can find them.
Buoyancy Control Device (BCD)
My first BCD was used. Worked just fine. Make sure whatever you can buy can be serviced at your local dive shop. And for heaven sake...have it serviced before you dive it. A jacket BCD has more "complicated" parts than a Backplate/Wing, so it is unlikely you can service it yourself so this is a "Maybe". A Backplate & Wing (BPW) is something you can pickup on the used market, just make sure the wing itself is in good shape as that is the bulk of the cost of the setup.
Weights
It's lead. Literally...if you can find some used..absolutely buy it.
Compass
On the low end of the market there isn't much of a reason to buy used. However, if you can find a Suunto SK8 that is used and in working order...buy it.
Tanks
High risk when buying used. If possible, require a recent Hydro at a minimum before you purchase or factor that into the cost and if it fails...now what? I got my 4 tanks used for a screaming deal...but I took the risk they would fail Hydro.
Drysuit
It's hard to say "No" for this because of how expensive they are, but I really want to. A high quality drysuit is generally a custom or semi-custom fitting. The odds of you finding a great fit and in good condition is going to be rare. But if you do find one...have it serviced / leak checked before you dive it.