If you’re really interested in learning about photography, a compact/point and shoot isn’t going to be too satisfying. They have a very limited ability to play with depth of field, which is one of the main tools you have as a photographer. Most compacts don’t even have real apertures, just a drop-in neutral density filter to decrease the exposure.
Two of your requirements (night shots and action shots) aren’t very well suited to compacts either. I’ve taken a few good action shots with compacts, but it’s so much harder than with DSLRs. Compacts are also very limiting when using low light. They simply don’t gather all that much light, their sensor is about 1/20th as large as the APS-C mainstream DSLR format and while they do make fast (wide aperture) lenses for compacts, they typically sacrifice zoom range. DSLRs of course make no sacrifices like that, since you can always find a lens that suits your purpose.
If you do want to get a compact, I think Canon makes the best one. I’m not sure if there’ve been replacements to these cameras in the last few months, but the S100 is great if you want something with a limited zoom range that takes high quality pictures and has decent low light performance, and the SX260 is great if you want a very wide zoom range with pretty good performance all around.
For DSLRs, I’m the Pentax guy around here. They simply build flat-out better cameras for the price, and they’re really oriented towards photographers and advance amateurs in terms of ergonomics and features and design. They’re also the only company that doesn’t artificially cripple their mainstream cameras in an effort to try to get you to upgrade to a more expensive camera to get certain features.
In particular, the K-5 with kit lens is a crazy cheap $769 on Amazon now. It was the best APS-C/mainstream camera there was for about 2 years, only recently having competition from new models. But you’re essentially getting a $1500-class semi-pro model with the full range of features for just a couple hundred bucks more than crippled entry level cameras.
Compact (Canon and Nikon deliberately add empty space to their cameras - seriously - because there’s a perception that bigger cameras are more professional), good ergonomics, fantastic sensor, very durable (stainless steel and magnesium alloy, doesn’t feel like a cheap plastic toy, full weather sealing - I’ve had mine covered in mud and used it in the middle of violent storms), pro-class features, 14-bit raws, 7.5 frames per second, in-body stabilization (this isn’t an unambiguous plus because in-lens stabilization can have a slight advantage in certain scenarios, but it does mean that every lens is stabilized whereas with Canon/Nikon it depends on the lens and makes the lenses more expensive). The only real disadvantage is inferior tracking in continuous focus mode, but that’s compared to the semi-pro and pro Nikon/Canon bodies, it’s still better than the lower tier cameras in your budget. Probably the best deal in photography history, at least in the digital age. Combined with the best-in-class 55-300 you’d have 18-300 covered to begin at just $100 more than your current budget. I’ve taken most of the top third of these pictureswith that combo. My second recommendation would probably be a Nikon D5200, but since they’re about the same price, and the K5 is way more camera, there’s no reason. You’d have to go down a tier to the D3200 level cameras to save money, and those are several tiers below.
Incidentally, everyone will tell you that the body isn’t that important, that it’s the lenses that are the important thing, and no, I’m apparently the only person to disagree with that. Yes, better lenses take better pictures, but we’ve reached a point where even the cheapest lenses give pretty good results. Whereas a body that fits you, that you feel comfortable taking with you, that has a great sensor (and sensor performance between cameras can be just as big a thing as the difference between lenses with similar purpose but different price tiers), that fits your style and gives you all the features you want is more important to getting you out there and learning and mastering the art of photography. Start with a good body, and you can build up your lens collection over time. I’d rather have a semi-pro or pro-class body with $100-200 primes than an entry level body with great lenses.
As far as software goes, you should shoot raw (the camera records exactly how the light hit the sensor with no processing on-board from the camera, which gives post processing software the most information to work with) and use Lightroom. I’m not sure photoshop is really necessary at all. Depends what sort of editing you do - if you just want to tweak stuff like contrast, exposure, white balance, levels, saturation, etc. then Lightroom is better. Photoshop is really only useful if you’re trying to do advanced stuff like adding and removing elements from pictures, altering the way things look, etc. I saw Lightroom 4 on sale for $50 not long ago, but I don’t know how often something like that comes along.