OK. First a disclaimer.
In the world of interchangeable lens cameras such as the Nikon D5500 and D7100, there are 2 main camps. DSLR & Mirrorless. DSLR cameras have a mirror which is used for autofocus while mirrorless cameras do not. Mirrorless cameras have the AF directly on the sensor (the rectangular blue thing in the lens mount below). The advantage of the mirror (DSLR) is faster and more accurate Autofocus (especially when shooting rapid continuous); the disadvantage is added size and weight.
I was an early adapter of mirrorless APS-C cameras when I bought the Sony NEX-3 in 2011. I did so because I wanted a small easy to carry camera that did not sacrifice image quality. The point of all this is that I do have a bias towards mirrorless cameras. In 2013 I purchased another mirrorless (NEX-7) and just recently I purchased my third (Sony A7rii). I do believe that the future of cameras lies in the mirrorless design. The first few generations still lagged behind DSLRs in terms of Autofocus, but the more recent ones have closed the gap. In fact, unless you are a pro-sports photographer, the differences are negligible. So again, I do have a bias towards Mirrorless cameras.
Having said that, the Nikon D7100 and D5500 are fantastic cameras. The D5500 is right in your price range and would give you very good image quality, ability to shoot terrific action shots and with the right lens shoot good shots in low light as well as portraits. You could buy the camera with the kit zoom for under $500 and get a solid prime or two and still be at around $1,000. (If you don't know the difference between a zoom and a prime lens, don't panic. Neither did I when I started). I'm not a big fan of the "package" deals you pointed out. It includes a lot of "cheap" throw-ins to make it sound like a good deal. The tripod is likely either very small or not of high quality. You are much better off saving for a good tripod.
Lets compare the D5500 to the mirrorless Sony A6000 (btw, Fuji also makes great mirrorless cameras). For under $500 you can get the A6000 with the kit 16-50mm collapsible zoom. Image quality and all the other things are basically equal to the D5500. But the camera is significantly smaller. This is a top view - note that the A6000 has the kit lens attached while the D5500 is just the camera body.
Here is a good comparison of all the camera's features and specs...
http://cameradecision.com/compare/Sony-Alpha-a6000-vs-Nikon-D5500
Another big difference between a DSLR and mirrorless is the Viewfinder. DSLR use "optical" viewfinders which use a prism to show you a view close to what you see with the naked eye. Mirrorless uses an Electronic View Finder (EVF) which is a digital screen image of the view. Optical viewfinders (OVF) are generally brighter and easier to view details, Electronic Viewfinders however give you a "WYSIWYG" type image. If you make changes with your camera controls, the EVF view will also change. This is very helpful when you are shooting in different and changing conditions and for me is another reason I prefer mirrorless. At Disney you can go from very bright sunshine to a dark ride quickly. Other appreciate the advantages of an OVF. I think as someone coming up from point & shoot an EVF would be more comfortable and easier. Plus is a great training tool. When I started I would look through the EVF and observe how changing Shutter speed, Aperture and ISO changed my exposure in real time.
Back to our comparisons. After you buy the camera and kit lens, differences in cost of other lenses start to become noticeable. In general, mirrorless lenses do cost more than their DSLR counterparts. A Nikon 35mm 1.8 DX lens will run you less than $200 while the Sony SEL 35mm 1.8 costs about $450. The Sony lens does have stabilization ( keeps the lens stable which is helpful in low light) but cost difference is still significant. It really comes down to Nikon (and Canon) have sold many more lenses than Sony and they currently have economies of scale. A DSLR 50mm 1.8 lens can probably be had for $175 vs. $250 for the stabilized Sony SEL 50mm 1.8. Something to consider.
In addition, the size difference advantage doesn't really carry forward to the lenses. With some exceptions, a mirrorless lens will be roughly the same size as it's DSLR counterpart all things considered ( although mirrorless lenses can be designed smaller but that's a whole other discussion ). But - the A6000 with the kit lens is almost pocketable. The D5500 is not even with it's smallest available lens. Mirrorless is much easier to carry around and a great option as you run through the parks.
Confused yet?
Another major difference. Because a mirrorless has to power an EVF, battery life is lower than an DSLR. I usually carried an extra battery when I hit the parks - the good news is they are the size of a box of tic-tacks, but the difference is worth noting.
So let's now compare costs for a good "starter kit"
For $1,000 you could get
Nikon D5500, 18-55 zoom lens (good for daytime shots), 35mm 1.8 prime lens (good for low light), 50mm 1.8 prime ( low light and good for portraits), a decent tripod or detachable flash.
or
Sony A6000, 16-50 power zoom lens (daytime), 35mm 1.8 prime lens.
or
Sony A6000, 16-50 power zoom lens, 50mm 1.8 prime, decent tripod and detachable flash.
Now that we said all that, it's important to mention a few other options.
1) Canon entry level DSLR and lenses ( similar in cost and quality to Nikon).
2) Advanced/premium "point & shoot" cameras. There are high end point & shoot cameras that in many cases will perform just as well as a DSLR or mirrorless with a kit lens. It would be a disservice not to mention those and explain them which I'll have to do later (or others may chime in). It also may be a solid option if you are not sure about expanding your photography past getting better family pictures and travel photos.
3) Micro 4/3 cameras. Also mirrorless but with a smaller sensor. They are known for being responsive and easy to use, good image quality and small and light body and lenses. I'm not an expert in those but can get into comparing.
Last and not least - Education. There are a lot of resources online. Check "digital photography school" for one. Also some good books. I read "Understanding Exposure" by Bryan Peterson but there are others. What you want to understand first is the "Exposure Triangle". This is the relationship between the 3 variable you can control to manipulate your exposure (shutter speed, aperture and ISO) and how they interact with each other. After that, you'll learn how to use those variables in a creative way.
More to come. I do hope others also answer. There are some great photographers on this site that can offer different perspectives (even some that don't like mirrorless cameras
). Not just camera commercials, but also how to use a better tool to become a better photographer.