This is to be expected. The mode you are using when the rear screen shows the shot is called "live view", and in this mode, the camera uses contrast-detect autofocus, which is very accurate but inherently slow. If you use manual focus or prefocus, the lag should be somewhat reduced.
In "normal" SLR mode, the mirror directs part of the light to your viewfinder, and some of it to autofocus sensors located at the bottom of the camera body. The sensor is totally obscured. With live view, the mirror is lifted out of the way and the main sensor used.
The double sound you hear is the mirror flipping down and back again. I haven't used the D5300, but I found others complaining about it. A post there explains that in Nikon's consumer-level cameras, the mirror flip is inherently linked to the mechanism which closes down the aperture — in both modes, the focusing is done with the lens at its widest aperture setting to let in more light.
So, you can't manually focus to avoid this, and metering isn't the problem either. (This same complaint occurs in many cameras in burst mode with manual focus and manual exposure — there seems to be no reason to cycle the mirror, but it's common to do so anyway.)
Some cameras are much faster in live view. Sony's "SLT" design leaves the mirror in place all the time, using the partial split to feed the autofocus sensors and the main sensor, not a viewfinder. Other cameras have tiny phase-detect sensors mixed in with the pixels on the main sensor. And maybe some avoid the mirror-flip — although at this point in time, it's very common for it to happen anyway.
The best advice, I think, in your situation is to use the viewfinder. You'll still get the mirror flip, but just the once — and that's the sound most people think of as the shutter.