Something the other answers seem to have missed and I will focus on is:
would "it" (the Sun) continue to travel through space after being knocked out of our system
Let's ignore the impracticalities of moving the Sun and focus on what would happen once the Sun starts moving. The answer is also not a lot would happen.
For a start, our Sun is already moving, along with the rest our solar system, at 230 km/s around the "Galactic Central Point". This "Galactic Central Point" is the galaxy's centre of mass, just like the moons of Jupiter, or the planets of our Sun, all stars in our galaxy are orbiting this centre of mass.
So when we talk about moving the Sun, what we really mean is significantly altering its orbital speed. This would change our path around the galaxy, but the consequences of that may not be seen for millions of years and are therefore difficult to predict. But due too the vast emptiness of interstellar space, there is a high chance nothing would happen.
That is if you ignore the effects of something large enough to change the Suns course in the first place. As has already been pointed out by other answers, you need something massive to influence the Sun.
Jupiter is the second largest object in our solar system, its mass alone is 2.5 times larger than the rest of our solar system combined. It's so massive that it forces our solar system's centre of mass outside of the Sun, if only slightly (Our moon has a similar, but much smaller effect on us). But as massive as Jupiter is, slamming it into the Sun will hardly affect its orbital velocity (as already pointed out).
So we're going to need something bigger than the biggest planet in the solar system and this is where we run into tangible problems for Earth. As I already pointed out, Jupiter is so big that it moves the centre of the solar system, anything that big entering the inner solar system, on a direct collision course with the Sun, is much more likely to disrupt our orbit before it disrupts the Sun.
This new gravity source could alter our path, taking us out of the habitable zone, perhaps making the planet too hot or cold too support life for much longer. It could even elongate our orbit enough to put our highest point into the asteroid belt where the planets surface would be bombarded by asteroids. Worst case Earth could even be accelerated out of the solar system.
In short if anything big enough to affect the Sun ever comes that close to the Sun, we are probably already dead. Where the Sun goes, the rest of the solar system follows, unless another force disrupts the planets orbits.