If the Sun's trajectory were altered, would the solar system follow?

The motion of bodies orbiting anything, is described by orbital elements - their distance from the barycenter (central mass), direction, velocity - and so two bodies of mass significantly lower than the central mass, with identical orbital elements, will move along the same orbit; with very similar elements their orbits will be very similar. For example, an astronaut on ISS, will circle Earth in nearly perfectly same trajectory as the station itself; both the astronaut and the station are vastly lighter than Earth, they are at the same altitude, the same speed, and so the astronaut is in freefall inside the station, not pushed against the walls or otherwise accelerating relative to the station - they both orbit Earth, their trajectory mostly unaffected by the minuscule differences in location between center of mass of the station and the astronaut - that difference being of order on ~30 meters at most, while the station orbits Earth on an orbit of radius of about 6700km (Earth radius + 400km) - with such small difference in orbital parameters they are both in the same orbit.

Similarly, Sun is about 25,000 light years from the center of the galaxy, and as you can guess the galaxy is much, much heavier than the Sun. Pluto is about 4.6 light hours away from Sun, so it moves 'in' and 'out' relative to the galaxy by maybe 9.2 light hours. So the influence of the galaxy on it practically doesn't differ between these positions - the galaxy influences it just the same as it does the Sun, so they both orbit the center of Galaxy on practically the same orbit.

Now, there is a certain rather significant difference between the orbit described by the Sun, and, say, Mercury. The Sun moves in the orbit around the galaxy at 230km per second. Mercury circles the Sun at 47.36 km/s. Now, when it goes "in the opposite direction" relative to the Sun than the motion around galaxy, that's 230-47=183km/s; when it goes "in the same direction", it's 230+47=277km/s; a body moving at either of these velocities at the same 25,000km from the center of the galaxy would have a much different orbit than the Sun. But as Sun's gravity reverses Mercury's direction relative to the center of the galaxy every 44 days (88 days is its orbital period), it all averages out over the 226 million years to complete the orbit around Milky Way.

So, in short, yes, the entire solar system is affected just the same as the Sun and orbits the center of the galaxy just the same, all the differences either averaging out or too small to be noticeable. And if you tried to change Sun's trajectory by a gravitational influence that is sufficiently strong and sufficiently distant, you'd redirect the entire solar system just fine. Of course if you used a source of gravity that varies more across the width of the solar system - say, another star or a regular (not supermassive) black hole - you'd completely break the precarious balance of orbital motion of our system, planets closer to the new source of gravity captured by it and entering its orbit, leaving the orbit of the Sun, planets farther out possibly escaping into interstellar space.

Gravitational field from a central mass (like a star, or the center of galaxy) varies with square of the distance from the central mass, so you need it acting over relatively short distance in at a very long distance from that mass, to have it "locally" behave like uniform field and act on all the affected objects nearly the same.