No difference in shooting. There's a setting you need to change.
On your 5100, it should be under the "Shooting Menu", the one with the icon that looks like a camera. Use your navigation control to go down to "Image Quality" and hit "OK". This will bring up a list of selections for how you want the image saved. You'll want RAW+F. After selecting that just exit the menu.
Now whenever you shoot it'll save two files for each picture. The jpg, which you were already getting plus the RAW file.
Nikon has a free correction software called NX studio that you can download and make these corrections. There is a slight learning curve to it, but the cool part is, unless you purposely tell it to, the software will keep track of any corrections you've made and if you don't like them you can simply undo them. In other words, it doesn't permanently change the RAW file. You can always go back to square one and redo the changes any way you want them. If you play with it a little bit, you'll get the hang of it pretty quickly.
If you don't like that one, there are also free trials of Adobe Lightroom or DxO that you can try as well.
I have Adobe for work, but I ended up buying a copy of DxO because I liked some of the additional features like their "denoising" module. That feature can remove a lot of that soft white haze or as many pros call it "softness" that comes from shooting with a high aperture, high ISO, and usually at the full extension of a tele lens.
If you're doing portraits of people, you can also use the correction software to "retouch" and heal photos if somebody has a blemish they don't want in the picture or whatever.
One of the things I also liked with DxO is that they are the company who tests almost every camera companies equipment and verifies all of their specifications. In doing that, they test camera bodies with a variety of lenses. So, whenever I upload a picture with a different camera body and lens combination the software will automatically correct for the distortions that they have found in that combination. It's crazy what they can do now.
In fact I saved your pictures to my computer and opened them in DxO and it automatically detected that they were taken with the D5100 w a 100-300mm lens. It's not as pronounced as the results from a RAW image, but here's your butterfly picture with corrections from the jpg:
Your original image:
Great shot by the way!
And here's the slightly corrected version:
Not a huge change but a little more vibrant colors.