First of all, welcome aboard and good luck in your beginning journey with photography. It can become a a rewarding and lifelong passion if you stick with it.
The thing to remember at this stage since you are just starting out is that any worrying over this camera or that one is really not necessary. Your current skill level, or lack of it, will level the playing field for any cameras you are considering. Each of the models that myself or others here recommend in this thread will be possess capabilities far beyond your current ability to exploit, and will still have plenty of room to grow into in the years to come. In other word you wont be out shooting any of these cameras any time soon. Heck, I have been shooting for almost ten years and my Sony a7 is still a better camera then I am as a photographer.
With that in mind it will come down to ergonomics, cost and lens selections. Here is a basic rundown of your major options with some generic info. I will also include used options as well as new since buying a pre owned camera makes a lot of sense for a new shooter. It can save you money on a body which leaves you more cash for lenses (its all the about the lenses, bodies come and go) and can reduce your investment if you choose not to stick with photography.
KEH.com is a great place to buy used with a no questions asked money back return policy in the first 14 or 30 days (cant remember exactly.)
FUJI- Great cameras with a absolutely wonderful lens selection, especially affordable prime lenses. All their cameras are APSC sized sensors, which means they are the biggest you can get in a mirrorless short of going Full Frame. Some models to look at here are the new
XT-10, a fairly small and retro SLR styled body with many of the abilities of the top of the line XT-1 with only a few corners cut here and there. You can probably squeeze out one of these with a single lens for around a grand, maybe a tad more. For used options there are the XE-1, XE-2 or even the X-Pro 1, which still might be found new for very little money since the X-Pro 2 is right around the corner. These three options are styled like classic rangefinders instead of like a traditional SLR. My only issue with the XE's and X-Pro is that they don't have tilting back screens. But if this isn't an problem they are excellent cameras and can be found used relatively cheaply. Pop on the new Fujinon 35/2 prime lens on any of these bodies and you have a wonderfully compact, powerful imaging tool. Oh, some of the Fuji's come with great weather sealing and some of their lenses do as well.
SONY- As it has already been said the
a6000 is one of the best deals going right now in APSC sized cameras. It is a stupidly capable camera for very little money. One issue with Sony (although this is changing) is their lens lineup isn't as fleshed out or as well thought out as some of the others. But don't let this stop you as they should have focal lengths to cover most of what you would ever shoot. And as with any mirrorless camera you can adapt old film lenses to use on this for some wonderful results. Please take a look at the following flickr folder of mine. It is from the older Sony NEX 7 of which the a6000 is the successor. All of these photos were taken with adapted film lenses. This type of photography is not for everyone but it is an option if you are so inclined. But it will give you an idea of what is capable with the a6000.
NEX-7
The ergonomics of the Sonys are quite different from the Fuji. They have a more modern design with a larger grip and have very little 'retro' styling about them. The Fuji's on the other hand look and handle exactly like old film SLR's and rangefinders. Myself, I prefer the older style especially since it usually places all the buttons and dials on the body that one normally controls the basic camera functions with, like shutter, aperture and ISO. One thing to note is that most Fuji lenses have an actual aperture ring on the lens, which is hard to find in this day and age and a welcomed feature. On the Sony you control the aperture on the body.
MICRO FOUR THIRDS- These are cameras put out by Panasonic and Olympus and they share a sensor size and mount so that lenses are interchangeable between brands. Keep in mind that the sensor is a bit smaller then the APSC sized cameras listed above and this can have some effect on the image size and noise. Of course, having said that, my local camera store has photos in their shop several feet in length that were shot on M4/3 and they printed them this large just to show how big of a print can be gotten from such a small sensor. I don't personally know a whole lot about individual models of this type, but I can tell you they have an absolutely GREAT lens lineup and most of the bodies have great weather sealing built in. They are the camera line that stays truest to the mantra that a mirrorless camera should be small and portable. The bodies from Panasonic and Olympus come in both modern and traditional (read retro) styles so there are many options to choose from. Would I personally shoot with one of these cameras? No, but that is due to my adaptation of film lenses and my love of bokeh. The smaller a sensor is the more it impacts the 'apparent' focal length of adapted lenses (field of view) and it also effects the amount of depth of field (bokeh) at a given aperture. For instance, my Canon FL 55/1.2 gives me a truly narrow DOF on my Sony a7 because it is a Full Frame body and the lens stays a 55mm. On a M4/3 body this same lens would have the field of view of a 110mm lens due to the crop factor and also have the equivelnt aperture of between f/2 to f/2.8, so I wouldn't get that super creamy background I love so much. This isn't going to be a huge thing for you since you are starting out, but just something to consider since if you end up sticking with photography then the decisions you make now can effect you options down the road depending on what you end up shooting.
Well that's it in a nutshell. There is a lot of information to consider when getting into photography but just remember what I said earlier. Any of these cameras and lenses are capable of world class images and what you get out of them will be far more dependent on your skill as a photographer as opposed to individual features of any given camera. A competent photographer should be able to pick up any camera and make compelling and beautiful imagery. But it takes years of shooting and a good understanding of the basics to get there. Photography is like anything else and it requires practice, practice, practice to find your 'eye' and to 'get good'. Hardly anybody picks up a guitar and is immediately able to join a band and start playing beautiful music. It takes dedication and a love of the art to stick with it. Photography is no different.
Hope this helps.