If you use Google Chrome, it's fairly simple. (There are probably easier and faster ways to do this, but it's what I've figured out on my own.) All you have to do for Instaposts that have only one photo in them is hit Ctrl+U (or alternatively, right click somewhere on the post and select View page source), which will open a new tab showing you the HTML source code for that post. Then hit Ctrl+F, search for ".jpg", and you'll find the original source image for the post. So, as a demonstration, as it shows up on my computer:
Here's the link to the post: https://www.instagram.com/p/BjH37tulxxn/
Hit Ctrl+U once you've gone to it. Once you've got the HTML source code, search for ".jpg", and you'll find the first result takes you to line 161, which reads:
<meta property="og:image" content="https://scontent-ort2-1.cdninstagram.com/vp/0ca9d8893d1c97d7415e81df5cef4f2f/5BA72FD2/t51.2885-15/e35/32144321_873466709507669_622870395082506240_n.jpg" />
The link in there is the original image file; copy+paste it and there you have it: https://scontent-ort2-1.cdninstagram.com/vp/0ca9d8893d1c97d7415e81df5cef4f2f/5BA72FD2/t51.2885-15/e35/32144321_873466709507669_622870395082506240_n.jpg
For posts that have multiple images that you swipe through to view, it's a little more convoluted (for me, anyhow); for these, I hit Ctrl+Shift+I, which opens up the developer tools on the right hand of the screen, while the page shows up on the left. (On my computer it defaults this to a mobile preview instead of a desktop, but it all works the same.) There should be a line of mini-tabs in this developer's tools window, going like so: Elements, Console, Sources, Network, Performance, etc. Click on Network; you'll get another subrow of tabs listing the various types of aspects of source code and whatnot for the page; so if you're looking for the images on the Instapost, click Img (click Media if you're trying to grab videos off of Instagram posts/Stories), and a list of the images viewed so far will show up, along with a preview window to the right of the list. You're still able to click/swipe through the other images of the post in the preview window of the page on the left of the screen, and as you click through, the list on the right will add them, and you can click on them there and see the preview, and open the original file in a new tab.
Hopefully that last bit wasn't too much of a muddle and you're able to figure it out well enough. And if anyone knows of simpler ways to do these things, feel free to sound off, as I'm aware of how clunky are the methods I described just above.