I've noticed I only take 2 types of pictures, action and nature. Should I try to diversify what I take? Am I being amateurish by just taking those "wow" shots?
It's likely that your best photos will always be those you take of something you are interested in. Don't diversify for its own sake. If you want to, do it. If not, don't. Some people extol the virtues of mastering a particular style/subject/piece of equipment/focal length/etc. as a path to the best photos possible, others are more interested in variety as a stimulus for broad based knowledge, creativity, or enjoyment. It's really an individual choice. Make photos of stuff you like photographing, make photographs to get better at photographing, make photographs to make good photographs - just do whatever suits your goals/desires.
I am looking for some criticism on how I can make my photo taking ability better. Should I start exploiting the settings of my camera? I would like to eventually move up to a DSLR, but I do not know if I should practice more on this camera or just go ahead and learn with a DSLR.
To get better (in general), you need to develop both your technical and compositional skills as well as any skills peripherially related to your photography (i.e. woodcraft and patience for nature photography, working with models/people if you want to photograph them, etc.). The balance of what you need to develop depends a lot on what you want to shoot, what tools you have/want/can acquire, and what your current strengths and weaknesses are. The skills you use when shooting different types of photos, shooting with different equipment, and shooting with different styles are all different. The compositional part is often the hardest to develop and the most difficult to communicate except by non-ironclad rules of thumb.
A few overriding technical differences between something like your camera and something like a DSLR are worth mentioning right now. The biggest advantages a DSLR or similar platform have over something like you have are going to be related to speed of operation and size of sensor. Depending on what and how you like to shoot, the availability of various lenses can matter a lot or not a ton. The biggest disadvantages are going to be cost and weight. My SLR kit probably includes 15 items that all have a retail cost that's as much or (significantly) more than the majority of point and shoot cameras. Most of them cost significantly more than your current camera. Packed into my camera backpack they weigh something like 35 pounds.
Returning to the fundamental advantages of speed and sensor size, these come into play in a few main ways:
Manual control of an SLR is both easier and more effective. At worst you press a button and spin a knob/dial. Most of the time you just spin a dial. That's infinitely better than the alternative button-mashing of point and shoot type cameras.
DSLRs tend to put out better-quality images, especially at higher ISO sensitivities due to their larger sensor.
Due to the difference in sensor size, changing the aperture on DSLRs has strong effectson the image
while for anything other than extreme closeups it doesn't really do anything on compacts other than make you change shutter speeds. On an SLR changing the aperture dramatically changes how much of the shot is in focus. This can be used to blur backgrounds or draw attention to specific details. You can't really do this with compacts. If you do mostly landscapes or otherwise want everything in focus this isn't a significant advantage. For other stuff it's often huge.
If I should go with the DSLR route, how much should I be spending for an entry-level camera. Would $2,000 be enough for the body/lens/numberous accessories? (If this is so I have some saving up to do
That price can certainly get you an SLR setup. The extent of what constitutes a relatively complete system differs from person to person depending on needs and budget. If your goal is high-level wildlife or sports photography that budget wouldn't even accommodate many lenses you'd probably want by themselves. For less equipment-intensive photography it could be stretched a pretty decent distance. Don't buy lenses/accessories/bodies to have them, buy them to fit specific needs. The price differences between bodies are strongly tied to features, not image quality. Features are great and often necessary, but be aware of what you're paying for and list out as much of what you want to be able to do with your budget as possible before trying to spend it. Spend money on equipment if you need to do so to solve a problem, not just to get equipment you perceive as nebulously better.
One last question. What photo editing software would be good for myself with little experience. Do I go straight for Photoshop/Photoshop Elements, or is a generic editing program fine?
Photoshop is great. It does a ton, packs a lot of power, and is smooth to learn when you know it. It's not necessarily newbie-friendly though. Lightroom incorporates an awful lot of what photoshop can do for photographers, short of retouching and similar. The most fundamental tools to have are the ability to adjust levels and curves, preferably in a manner that you can undo easily. Minor adjustments with those two tools are going to give you by far the most "bang for your buck" in terms of making images "better" without doing much to them.