What will happen if the solar system leaves the milky way galaxy? Is the milky way galaxy meaningful to us? Will any bad thing happen if the solar system just leaves the galaxy?
 A: In the short term, no, nothing bad will happen. The Milky Way is not particularly important to us in terms of keeping the solar system together, protecting us from anything dangerous, and so on. In fact it's probably slightly safer to leave:


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*If the solar system is leaving the galaxy by natural causes, chances are we're a few billion years in the future and the Milky Way is colliding with Andromeda, and our star is one of those that happened to get gravitationally scattered/ejected from the system.

*This is actually probably for the best. At present there are no known nearby stars that are likely to explode as supernovae any time soon. A galaxy collision typically causes the formation of quite a large number of new stars, many of which will be the short lived hot type that go boom. If one happened to form fairly close to the Sun and we waited maybe 10 million years, this could be very bad for Earth.

*On the other hand the Sun is expected to burn out around the same time as the Milky Way collides with Andromeda, so being in the galaxy might be advantageous if we're still around... we might want to find a new home around another star. There are many more options in the galaxy than in the void of intergalactic space.


Other consequences of leaving the Milky Way:


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*Some astronomers will be very happy. The Milky Way will no longer be in the way of a big chunk of the sky, making life much easier for them (I fall roughly into this category).

*Some other astronomers will be very annoyed, since they were trying to study the Milky Way and now it's getting more difficult to do so.

*The night sky will get a lot less spectacular, especially in the southern hemisphere (where you can look out toward the centre of the Milky Way). On the bright side, the Milky Way-Andromeda collision will be quite the firework show, so at least we'd get to see that before the view gets a bit more boring.


In response to comment on potential harmful radiation sources:
Much of the astronomically sourced radiation that is potentially dangerous to humans is actually from our own galaxy. All galaxies are dumping radiation into space, but ours being nearby is the dominant source. For instance, here is the sky in the ultraviolet (GALEX satellite); the orange is higher intensity (more info/maps here):

In X-rays (ROSAT satellite); most of the energy is coming from the blue/green regions:

And $\gamma$-rays (Fermi satellite); red/yellow = higher intensity:

As you can see, there's one feature that is qualitatively similar in every map - there's a big band of emission horizontally across the middle. The images are oriented so that the galactic plane lies across the middle horizontally; our galaxy (and all galaxies) put out a lot of radiation. Rather than shielding us, being in a galaxy actually irradiates us more. It's a good thing our atmosphere does a pretty good job of attenuating radiation in most harmful wavelengths!
Of course in many wavelengths, the night sky is of secondary concern beside the Sun - which definitely puts out WAY more in the IR, optical and UV (you don't get a sunburn at night, after all), and has significant x-ray emission. I think the rest of the sky wins in gamma rays since the Sun doesn't have processes energetic enough to produce any. For the longer wavelengths (sub-mm/microwave/radio) I'm not sure off the top of my head whether most of the energy we receive is solar, galactic or cosmic in origin.
