Recommended camera for beginning hobbyist around $500?

My girlfriend and I are looking for a camera that we can use to take pictures of family and vacations that won't have blur and would be a good camera to learn basic photography techniques (using the cameras settings rather than turning the dial and hitting click.) I used my father's Canon Rebel T5i and found that it took great pictures and might be considering that, but that is the only camera I've used sans my phone's camera and I know there are, or could be, better options. I've been reading reviews on the Rebel line (t3i, t5i, slr) as well as Nikon, specially the Nikon 3300. We're looking to spend in the $350 to $600 range. I've noticed bundles that include filters, cleaning kits, additional lenses, ect. So that could come into cost if worthy and pointed in the right direction but isn't important. Any input is most appreciated.

Examples of main use:
Football games
Kids playing in the yard
Scenic opportunities - mountains, wildlife, nature (we live in Colorado haha)
Vacation moments
Hanging out with friends
Eventually I would like to take a more artistic approach to those examples, hence fourth the move from camera phone to mid range budget.

Getting anything in price range from Canon, Nikon, Pentax, or Olympus will serve you well and you will most certainly be happy with anything from these guys. As has been said, take a look at Pentax K50 as it's the only one that is weather proof at this price range, and they have weather proof lenses too.
I would avoid any "kit" package that includes all of these filter, cleaning kits, etc… As these accessories are usually very, very cheap. Do yourself a favor and just buy the camera body and choose your own bag, cleaning kit (all you need is a lens cloth and avoid cleaning pens), a circular polarizer (tiffen, hoya) maybe a protective filter like a UV if feel it's needed, a cable release (buy the cheap off-brand) and that's about all you need to get going. Your options are typically limited to a kit from the camera manufacturer which includes a lens. If the body that you want is available body only, then I'd suggest getting it and choosing a lens better than the typical 18-50mm kit lenses as they're universally considered low quality.

The least expensive DSLR's are the Sony Alpha SLT-A58 - $450, Canon T3 - $460, Nikon D3100 - $480, and Pentax K-500 - $400. Any of these would be good to start with. Typically, bundles are junk. Just get the camera and kit lens, and worry about the rest of the stuff later. If you liked your father's T5i, but the $700 is out of your budget, consider the T3i, currently going for around $550.
http://www.bhphotovideo.com/...gital.html

I think that for the uses listed a mirrorless camera with APS-C sensor like the Sony A5000 would proabably suit you best. Its smaller size means you will feel less worried about taking it everywhere.
See.
http://www.bhphotovideo.com/...gital.html
If you look in the link below you can see the photos on flikr that people have been taking with it in the real world.

Bundles? Often put together to sell less desirable lenses with the camera.

In www.cameta.com ---the NIKON D5200 costs like less than $650 ( Free Shipping! )---the adjustable LCD screen is a plus feature--USA product! 24.1 megapixels-D movie format, Active D-lighting and so much more

I usually recommend against a DSLR camera such as the Canon Rebel series. Mainly because of the lenses needed. Such a a $850 for a macro, or $250 for a 50-200mm lens to extend your telephoto range reasonably, and maybe a portrait lens $? But you mentioned football games! Just how close will a 200 or 300mm lens bring in a football game? This extra long setting screams to me a quality bridge camera like the Pentax XG-1 with a 52x zoom lens or the Canon SX60 camera (I prefer Pentax brand). Your move from camera phone is to be applauded! BUT go slow!

An entry-level DSLR is a good place to start, but you'll end up spending a lot more on lenses, flashgun etc.

Find a dealer and see what Sony (better video), Pentax (the first Japanese SLR manufacturer, and the only way you'll afford a pentaprism), as well as Canon and Nikon have to offer, and choose the one that feels best in your hands.

While the camera is a useful tool, eliminating blur is down to you.