Moving from Bridge camera to DSLR does it make a big difference in image quality?

I currently have a Sony hx-300 from past 2 years. I'm planning to buy the nikon D5200 with dual lens kit. So would it make a big difference or should I continue with my bridge?

It will be like going from a golf cart to a real car. There's much, much more to camera than just image quality. But when going from a bridge camera to a DSLR, you will experience a huge increase not only in image quality but in the types of shots that you can capture, and capture well on a consistent basis.

As for image quality, what you will notice is that the images from an APS-C DSLR will have much, much less noise, much better color, contrast, dynamic range, and yes, sharpness. The sensors in DSLRs are much larger which means a few things. For one, larger sensor means larger pixels which is how you'll end up with less noise, better color, contrast and dynamic range. Secondly, the larger the sensor, the larger the image circle the lens has to project. The only way to achieve a larger image circle is to use a longer focal length. So while you've been using say a 15mm lens position on your HX-300, you will need to use an 80mm lens to achieve the same angle of view. Now, because depth of field is in part affected by the focal length, you will inherently get a much shallower depth of field. You can easily calculate the difference by multiplying the aperture used on your HX-300 by its crop factor, which is 5.6x. Shooting at f/2.8 with an APS-C with an 80mm lens will produce a shallow depth of field. But when shooting at 15mm at f/2.8 it's produces the same amount of DOF as an APS-C shot at 80mm with an aperture of f/16 (2.8 x 5.6 is roughly 16). Therefore you're going to be getting super-shallow depth of field relative to what you're used to getting with your HX-300 with it's 1/2.3" sensor. Blurry backgrounds and isolation of your subject will be extremely easy to do, while it's nearly impossible to do with your HX-300.

A major difference between a 1/2.3" format bridge camera and an APS-C format DSLR is the level of control. Because the massive depth of field with 1/2.3" format camera, you actually don't need f/stop smaller than f/8, which produces the same DOF on an APS-C camera shot at f/44! Therefore camera manufactures don't give you the option of using a smaller f/stop because it costs more to add a feature that isn't needed. But this prevents you from having more control over the exposure. Say, for example, you want to use a long shutter speed of 1/4 of a second to really blur motion. At f/8 at ISO 100, unless you're in very low light, your shutter speeds will be around 1/30th at f/8 at ISO 100. Now, with an APS-C camera, you will typically have apertures ranging from f/4 to as small as f/22, or f/32 on macro lenses. This allows you to shoot at much longer shutter speed and thus get the image that you want.

The D5200, as will any DSLR, focus much faster than any bridge camera. Its focusing system can easily track moving subjects which is something that a bridge camera like the HX300 can't do. The number of shots that the D5200 shoots per second is a mediocre 3fps (the less expensive Pentax K-50 shoots at 6fps). This number will drop when in continuous focusing mode. So if you're into sports, the D5200 is a poor choice.

Overall, you should be blown away with the results that you get with the D5200 or any DSLR for that matter. The level of control that you will have, and the speed at which the camera works will also be a huge improvement. So not only will all of your shots have better image quality, the types of shots that you can get (and get consistently well), will also be greatly increased over any bridge camera at any price. Of course, all of this increase in performance and capabilities is only good if you use it.