What's your budget, and what sort of transport do you have? Here's some commentary on individual components, not including camera and lens:
Three Speedlights - currently the 430EXII and two Yongnuo YN-560II. For most studio work, you don't need ETTL: In fact, ETTL can hinder more than help, since it may change the light output depending on the subject matter. The YN-560III is absolutely worth the price difference over the YN-560II, since the newest version includes a built-in receiver, and I believe the power can be adjusted remotely with certain triggers (niiiice).
The Godox/Neewer V850 and V860 also appear to be very capable contenders for Speedlights - when I wear out my YN560IIs I might switch over.
If you have a little more to spend, one or more studio strobes with modeling lamps can make a huge difference in precise portrait lighting. Downside, they either need an outlet or a big battery pack, they're heavier and harder to transport, and Speedlight modifiers may not work on studio strobes. I have entry-level Elinchrom studio strobes, but Paul C Buff products (Einsteins and Alienbees) are a good choice for entry-level studio lights.
2. Triggers. Things are rapidly changing here, so I'm a little reluctant to give specific recommendations. And, you might make your trigger selection after deciding on your lights. I have Cactus V5s, and while they were pretty good for the price 3 years ago, I don't think I would buy them again given the current tech.
Ideally, look for something that will allow you to remotely control the power of your lights; this will be such a huge convenience when your light is far away or on a tall stand. If you have Yongnuo YN-560II, I think there's a transmitter on the way that will remotely control power; right now the RF-603II will talk to the built-in receiver for triggering but not power control (I think). The Godox flashes don't have a built-in receiver, but you CAN control the power remotely with the FT-16 transmitter and receiver.
As an alternate/inexpensive solution, get an inexpensive trigger like the Cactus V5, a single receiver for one of the flashes, and put the rest of your flashes on optical slave mode. This is cheap and easy to set up, but if there are other flash users around, they'll mess you up.
The Elinchroms can be remotely controlled with the Skyport series triggers, except for the Skyport Eco (which is the one that I have, ugh) which is a simple trigger. Presumably, there's something out there that plays nice with the Einsteins and Alienbees.
3. Light modifiers.
Here's the big decision, and you can spend a lot or a little here. Many novices start with simple and inexpensive shoot-through umbrellas. For Speedlight portraits, I personally prefer the look of a brolly box or softbox over a shoot-through umbrella; the brolly box has less rapid falloff, less of a center hotspot, (IMO) more pleasant wrap light, and less uncontrolled spill.
A beauty dish is another option to consider; it gives a little bit different look, and is more robust for outdoor shots - a little less likely to fall over in a slight breeze. And though it's a little heavier than an umbrella, it's easier to carry around on a stick since it's not easily broken like the ribs of an umbrella.
Here are the brolly boxes that I use with Speedlights - after 3 years of hard use they need a replacement but they still mostly work:
4. Stands etc:
At a minimum, you'll want about one stand for each light, and the associated accessories to make other stuff work with them. Generally, the more you spend, the better you get - but there are some inexpensive gems out there. These are my starter light stands; they're now my backup stands - after 3 years, lots of travel, some spills and thrills, they're getting a little worse for wear, but I still use them often:
Taller and sturdier are better. I wouldn't go any shorter than 7'6".
To put Speedlights & umbrellas onto the stands, you'll need umbrella adapters.
If you have studio strobes, generally the strobe has a built-in mount to go onto a standard light stand, and the light modifier will mount directly to the strobe head, so you don't need adapters like that.
If you ever think that you might work outside, get some sandbags and USE THEM. If a breeze knocks your lightstand over onto a client, and the client gets clobbered in the eye with a light, you'll wish you had stayed home. Fill them with pea gravel - if they bust open or unzip, gravel is SO MUCH easier to clean up than sand.
6. Backdrop & backdrop stands.
I'll see if I can type up some about backdrops later.
White seamless paper should be in every studio photographer's arsenal. "Thunder Gray" paper is good too; it can be easily colored with gels or dropped to clipped black. Some heavy-duty cloth or canvas backdrops are nice, depending on what you shoot. By wary of cheap muslins, the ones that come in inexpensive kits are straight-up garbage.
There are some other must-have and nice-to-have accessories too. I love Manfrotto Super Clamps, but they're kinda pricey if you buy new. Still, get one if you can swing it, to clamp a Speedlight in just about any location.
Plastic spring clamps, cheap ones from Harbor Freight, get a sackful of assorted sizes.
Nylon cable ties or zip ties are great to have on hand.
Batteries! If you're using Speedlights and triggers that are powered by AAs or AAAs, Eneloop brand rechargeables almost can't be beat.
gotta go, I'll try to follow up more later.