When something is "in orbit", it experiences centripetal force. The centripetal force is provided by gravity, in which case whenever anything is "in orbit", it is "falling".
The sun is orbitting around the center of the galaxy, but not all things orbit the center of the galaxy. Our planets orbit around the sun, not the center of the galaxy. Why? They are in immediate proximity to the sun, whereas the galaxy is far away.
You may ask yourself, "if the galactic core is strong enough to pull the Sun in to an orbit, why aren't the planets affected?". Well like I said, our planets are closer to the Sun than the galactic core.
Here's how much force the galactic core exerts on the Sun:
F = m * a = G * ( M / r^2 ); F is force, m is the mass of the Sun, a is acceleration due to gravity, G is the gravitational constant, M is the mass of the galactic core, and r is the distance from the core to the Sun
F ~ ( 1.99 x 10^30 kg)( 6.67 x 10^-11 N * ( m / kg )^2 )( 7.36 x 10^30 kg )/[( 2.5 × 10^20 m )^2]
F ~ 1.563 x 10^10 N
Seems like a lot, doesn't it? Well, it is. But compare that with the force on Earth, for example, from the Sun:
F = m * a = G * ( M / r^2 ); F is force, m is the mass of the Earth, a is acceleration due to gravity, G is the gravitational constant, M is the mass of our Sun, and r is the distance from the Sun to Earth
F ~ ( 5.97 x 10^24 kg)( 6.67 x 10^-11 N * ( m / kg )^2 )( 1.99 × 10^30 kg )/[( 1.496 × 10^11 m )^2]
F ~ 3.54 x 10^22 N
As you can see, the force from the Sun is much more powerful on the Earth, then say, the galactic core on the Earth (even though I calculated the force from the Sun to the Galactic core, the trivial distance will result in an answer of the same magnitude).
The Sun and other stars are orbitting the Galactic core (which is just a bunch of stars, but I heard there was a black hole somewhere). The planets are orbitting the stars, and indirectly orbitting the Galactic core. However, the planets are "falling" towars the Sun, not the Galactic core. Any object within a star's solar system will fall towards the Sun, not the Galatic core. It is not impossible, however, to not be "falling" at all; that point is called a Lagrange point.
So yes, even though the Sun is in orbit around the Galactic core, it doesn't mean objects in every point of space is affeceted.