As photographers we can really get caught up in gear. It's so easy to believe our inability to improve is related to the equipment we own, when in reality the cheapest digital camera is in many ways better than high end cameras of yesteryear. We are limited only by our skills. I have lately forced myself, whenever I get bitten by the gear-bug, to instead of researching or pouring over online catalogs and reviews, to plan a photo shoot idea, and go out and shoot instead. Practice, a photo trip, or a new book (be it instructional, or inspirational) will do far more for our skills than a new piece of equipment.
That said, I want to lay out what I think the basics are, in case you don't have a kit yet at all, or in case you are wondering if there truly is a limiting factor on your current gear. This is assuming you shoot in a similar vein as this blog, mostly portraiture where you control the model placement and can zoom with your feet. None of this will apply to bird photographers or sports journalists.
A camera with wide, normal, and tele lens options.
A light source of some type with at least one constraining option (like a snoot) and one softening option (like an umbrella or softbox) and a way to trigger it.
Wide for me is 35mm, but some people like 24/28mm.
Normal is between 45-60
Tele is over 70, and for me is 85
The reasons for this is your composition. You can keep your subject the same size between wide/normal/tele and totally change what you see in the background. When you want a sense of location, you use wide or normal, and when you want isolation, or a very specific chunk of background, you use the tele, which will also enable closeup portraits without distorting the face.
The light is so that you can brighten portions of your subject, or bring a more pleasing light pattern for it. It's not necessary all the time, but having even a cheap LED panel can really open options.
If I were building a kit today from scratch with my current knowledge and preferences, here is what I would buy.
Perfect kit without waisting money:
Canon 5D I or II
35mm 1.4, or f/2
Elinchrom Quadra with silver umbrella large/small, softlighter II 60", and XXS softbox and grids, large reflector with 8 degree gridspot insert.
A more budget option would be to go with any decent crop body of any manufacturer with the tamron 17-50mm 2.8 (giving you from wide to tele of good quality and decent lens speed) with a shoe mount flash, or an alienbee setup with vagabond.
The cheapest route would be a Canon s90, and a cheap shoemount flash on a stand, with a white umbrella, and an optical trigger (set off by the s90's flash, just turn flash compensation down so it's as dim as possible)
Remember, we are in this to make photos. Gear can be a hobby on it's own, but then you're not in the business of making images to communicate, you're a collector. Focus on shooting more, pursuing ideas, and use the gear-bug to refocus your efforts on shooting.