Yes, shutter speed was the main problem. Flash or no flash, wider aperture or not, you need your shutter speed to be faster to freeze the action and capture moving people.
1/250 is probably a good
bare minimum for moving, dancing people if you're not using flash, and faster than that (maybe 1/800) would be much better. You would do this by being in Tv mode on the mode dial, and dialing till you see 1/250 or more as the shutter speed. The camera will take care of the ISO and Aperture then to get the exposure right for the shutter speed you tell it.
I don't know exactly how good the K30 does with higher ISOs (the 3200 that shot was at is already pretty high), but I would imagine you would be happier with a higher ISO shot (more grainy, maybe starting to get some color noise) where the action was frozen, as opposed to the blurry subjects you got there.
I would also consider underexposing some by setting the +/- to -1 or -1.3 if that will help you get a faster shutter speed. The picture will look dark on the camera screen, but you can adjust it on the computer, even in the most basic of software, including the one that comes built in to Windows.
The Fundamentals of Exposure is a good article to read if this is all a bit too technical.
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Finally, I'm not saying you need to spend money to solve this, but this kind of poorly lit indoor shot is exactly where having a fast prime lens (like a
50mm f1.7, a
50mm f1.8, or even
50mm f2.8, maybe a
40mm f2.8, or else a
35mm f2.0 or
35mm f2.4 which 35mm might be a little better of a focal length for you) in your kit
really helps. Those wide apertures (lower f number) let you set the shutter speed higher.
You could probably find a used autofocus 35 f2 under $250, and a 50 f1.7 or 50 f1.8 under $150, if you look on ebay, and maybe a little more from a camera shop with a nice return policy. Or check in the marketplace here, I see some of those lenses there for even better prices than I mentioned!