Gear is always a difficult question, and my general advice is to borrow/rent before you buy. Also, it very much depends on where you’re diving…here are my €0.02, though:
Mask:
I prefer alow volume mask, i.e. frameless (where the only solid parts of the mask are is the glass). I’ve seen frameless masks with peripheral glass, but I don’t see why bother. Frameless masks are also good to store in a pockets (if you want a spare).A large volume mask means that you have more water to press out if it gets filled, as well as more volume to fill to avoid mask squeeze (and more air to let out on the way up). Additionally you’ll get tunnel vision.
Masks are generally good to keep cheap, because they’ll at some point get stepped on/dropped/lost/fly off while the RIB is doing 40 knots.
The above advice about checking the seal is correct, and should really be the only important factor when choosing a mask, as well as making sure that it’s made out of tempered glass.
Alternate air source retainer
Ah, here we have a question of religion. I dive with a Hogarthian configuration
Fins:
A nation of people with a penchant for tango and cheap vodka. Also what you wear on your feet.
There are generally two types of fins, soft jetfin-type fins and hard fins.Which one is best is subjective, but generally hard fins give more power while soft fins give more control.
Personally I prefer soft fins, because I normally don’t want much power while diving (though I’ve swum against a strong current in mine with no issue) while control is vital. Unless you’re only going to be diving on sandy bottoms and far from any wrecks, you’ll at some point want to learn how to do frog kick and back kick.
Spring straps are not essential, but they lessen the risk of a strap breaking and make putting on your fins a lot easier.
Snorkel:
Buy the cheapest you can find. Mine’s never been in the water. You don’t need it under the surface, so why bring it?
Knife:
The classic image is of a diver carrying a foot long bowie knife on the inside of his shin…and what’s he going to do with that, stab a shark? 
A cutting instrument is very important to have, as you may get snagged in lines or nets. A small knife (without a point) or a pair of stainless steel scissors on your belt is best for the purpose and have a low stabbyness factor. A buddy of mine dives with a pair of garden shears, which just might be the best of both worlds.
Compass:
The simpler the better. Get a sturdy diving compass that you can mount on your wrist.
Light:
Not mentioned, but I’d keep my eye out for a good LED light. Good to have if you do a night dive in the future, for peeping into wrecks and for signaling. Get a light that you can use as a spare in case you in the future opt for a canister light.
BCD, Regs, Suit:
A suit is important that it fits and is relatively light weight, so I say buy one. It depends on what kinds of waters you dive in (at home, I dive in a dry suit year round). A common recommendation is to buy a 5mm suit for warmer waters, and a 7-5mm shortie which you can wear outside of your ordinary suit on chilly days.
With BCDs you can chose between the classic vest and wings. I like having my gear as simple and as maintainable as possible, so I always dive with wings (there is some religion involved here too).
Regulators are mostly equal, though not all regs are suited for cold waters, depth or nitrox. If you buy a regulator, at least make sure that it supports nitrox.
Diving as profession:
Note that professional diving is very different from recreational and technical diving. I’m not a professional diver myself, but I don’t think that a recreational certificate would count as much of a perk. But hey, just dive for the fun of it 