So what does a guy need to get his feet wet here? Vice, bobbin, scissors, whip finisher, and of course materials... hair, thread, cement, hooks or premade jig heads. What's the bare minimum equipment to start off with?
That's pretty much it, you've listed everything you really need. There's many good tying vises & tools on the market, and what's best is mostly personal choice & how much you want to spend.
Here's some tips for each. I won't try to recommend a lot of specific brands, as I can only tell you what I've used.
Vise - need to hold the range of hook sizes. Not all fly tying vises will hold larger hooks well, so make sure what you get will.
Scissors - Make sure they're comfortable to use. Most good fly tying scissors now have large finger loops, which helps make them comfortable to use. They should be sharp and fine points helps with cutting close & tight. If you'll primarily be tying jigs, with hairs, serrated blades are better than straight. A good pair will cost in the $10 +- range, if stainless steel. I use a couple of pairs of tungsten carbide scissors, which are in the $25 each range. A very good and popular brand now is Dr Slick, but there are others.
Thread Bobbin - there's many available. I highly recommend getting one with either a ceramic tube or ceramic insert and large enough to both fit your hand and wrap whatever you're tying. I've been using Griffin Enterprises bobbins since they started in business, but again there are other good brands. I like their Magnum size for tying jigs.
Whip Finisher - this is a tool many will recommend. It's not difficult to use, but not a "necessary" tool. I learned to make a whip finish knot with just my fingers, so don't use a tool. I bought a cheap one years ago, and never could get the hang of using it. There are some that folks will swear by! Materelli is a popular brand.
Threads - there are many sizes used in tying. For jigs, a good nylon thread is all you need. This might range in sizes from 3/0, A, B to T-70. Most are designated in denier now. The larger the number the heavier the thread. So, 200 denier will be heavier than 30 denier. I get some threads from fly tying materials suppliers, and some I buy at sewing shops in bulk spools. Basically the same in some types. There are also flat threads and round, twisted threads. Each can be useful. Round threads will build up faster. Being I've been tying flies for a long time, I started with threads from Danville Chenille Co, the oldest fly tying thread supplier in the USA, and still use their Flat Nylon Threads for a lot of tying I do. There are other very good brands now too.
Cement - there are "head cements" available, sold at fly shops for tying. A lot of us use Sally Hansen's Hard As Nails clear finger nail polish, which frankly is as good as any head cement. I also use some super glues and epoxies. They're generally used to coat threads to improve durability.
Materials - for jigs, there again are many. Really any hair can be used. There can be reasons to use specific types due to length, or texture as some will be finer or courser than others. I use many types. Materials like the rabbits strips can be obtained from suppliers of fly tying materials as can most other materials. I buy from many sources, simply to shop & get the best prices. If you hunt or have friends that hunt, that too can be a good source for some materials, especially deer tails. There are also some "flash" materials available and synthetic hairs that can be useful. I use such flash materials as "Flashabou, Krystal Flash, and Polar Flash" all being brand names. I like the Polar Flash the best.
Hooks or Jig Heads - if you want to get into pouring & painting your own, that's a whole different set of tools & supplies. There are various sources where bare or pre-painted jigs can be purchased. I pour my own, and those that I've bought from in the past include Jann's Netcraft, Lure Parts Online, Barlows Tackle and Terminal Tackle Co. These should all be listed in the pinned post at the top of the Tackle Box forum. There are also some sellers on ebay who have jigs for tying. There can be a wide range in prices here too, as much of the time the hooks used affects the price as does the painting process. I like "better" hooks, this being better than the older styles of basic jig hooks. I use Mustad Ultra Points, Gamakatsu, Owner, Matzuo, some Eagle Claw, Trokar, and Daiichi brands, plus some others. I powder paint all of mine, but not all makers do.
This should cover most of what you need! The rest will just be the learning process!