These days, I'd say you can't really go wrong with any DSLR on the market. All of them are good, solid cameras that will serve you well. I'm a Nikon shooter personally, but Canon's offerings all seem to be just as good.
The specific bits of equipment that I'd suggest for 1/6 shooting specifically (especially stop-motion) would be:
1. A good macro lens. You'll tend to shoot 1/6 stuff pretty close, and regular lenses may not be able to focus close enough for a 1/6 figure to fill the frame. If you're going with a compact-sensor Nikon model (they're the ones with 4-digit model numbers, like D3100 or D5200), there's a really nice 40mm macro lens (Nikon calls them "micro" lenses for some reason) that I shot with very fruitfully for a while. I've moved up to a full-frame Nikon camera (usually 3-digit model numbers, like D610 or D800; the exception is the newly released D500), and for 1/6 stuff I am now alternating between a Nikon 60mm macro (for full figure shots) and a Tokina 100mm macro (for real close-ups).
2. A lightbox. IMO, the single best thing you can do for your photos is make sure you have a good, consistent environment to shoot in. I started a thread documenting my DIY lightbox, and there are more threads in the general Photography forum here. I've actually moved up to a "Mark III" lightbox, where I dispensed with the box in favor of a frame made of PVC pipes (I think I used schedule 40 stuff, where pipes can stay in fittings without glue, which makes re-configuring the box or breaking it down for storage a snap). Should probably document that, but technically I'm not done with it.
3. A good tripod. You can skimp on the camera initially, since the starter models are still excellent, but DO NOT skimp on your tripod. Do it right and it will grow with you; do it wrong and you're just going to get annoyed and take crappy photos and and eventually spend more later to get a better one. You'll want a ball-joint head and good, solid legs. Common materials are aluminum and carbon fiber, with the latter being lighter and more expensive. Skip the ones you can find in department stores and places like Best Buy. They're crap. Manfrotto and Oben make some good, solid tripods that aren't too expensive (around $300-500 -- yes you want to be spending at least this much on a tripod) and will work real well with the smaller, lighter DSLRs out there. A good, solid tripod would be a must for stop-motion.
If you are on a budget, you can save money by picking up a used entry level DSLR (a basic Nikon D3xxx series is a fine starting point), or any of the manufacturer refurbished models (they carry factory warranty provided you buy from an authorized North American dealer). The one thing you want to be cautious, and I can't stress this enough, with when purchasing a DSLR is there are a great number of grey market cameras sold in the US which are Asian market models (models are the same, but serial numbers will denote the market) and carry no Nikon US warranty. These often carry dodgy 3rd party warranty or retailer specific warranty. The camera industry can be a very fly by night operation as far as retailers go (hello New York!) and some really dodgy stuff goes on. If you are new to cameras, a Nikon US refurbished model is a good budget option.
A basic kit lens (the ones included with most entry DSLR) will get you started or you can get a model (advertised as "body only") without a lens and pick up a macro lens as suggested as the quality of those lenses will be superior. The lens you use really depends on how much working distance you have between the item and your camera, as you are working with a fixed distance when using non-zoom lenses. This is key as you'll want to be using a tripod and it can get very awkward if you get the wrong focal length lens. The lens is ultimately the important bit to focus on buying, more so than the body for a beginner. Watch some youtube videos of demonstrations macro lens to get an idea of the working distances. You may find some are way too close to be practical for your use once you factoring in positioning a tripod.
For a tripod look at the Manfrotto 190 series with ball heads (as opposed to fluid heads which are more for video work, the 496 ball head is a common one you will see and ideal for a beginner). They are solid and versatile as you get adjustable height legs and adjustable height center column. Again another item you can pick up used at reasonable prices. Check Craigslist as you'll often find decent local deals from people upgrading their gear. If you want a trick setup, I paired one of these to a manfrotto 190 tripod with 496 ball head, using one of these. You'll have to be careful with the weight of the camera and lens setup, but this is an incredibly handy setup for figure photo taking. If you are using a tripod, get a remote shutter release (wired cords are fine), as you can throw off photos with pushing the shutter release on the camera as it introduces shake which throws off focus.
For lighting you'll want some sort of direct lighting setup for indoor use. Do not use on camera flash, and I wouldn't recommend off camera flash for a beginner given the cost and learning curve. You can dig through 4dEFCON's posts as he posted his improvised lighting setup for an affordable option. You don't have to spend a lot of money for a basic lighting setup. You have can have a $3000 camera setup but with a poor lighting setup for indoors, you might as well be shooting with a phone.
Last but least once you get going, you may find software like Lightroom very handy. It makes it a cinch to clean up photos, adjust exposure, and adjust color temperature, which makes the difference in the final photo.
It isn't cheap to get going, but if you buy the right gear first you'll save yourself a lot of money in the long run.
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