“So the speed at which the galaxy is moving is like keeping this one big fish bowl where we can swim from point a to point b without the solar system leaving us in its dust?”
This is fundamentally wrong. It’s not analogous to a fish bowl. The key is that there are no orbital mechanics in a fish bowl.
You don’t “swim” from a to b in outer space. All motion involves some sort of trajectory.
Satellites around the Earth stay in orbit around the Earth because they move sideways at high enough velocities.
I think that Newton’s Canonball is the best way of illustrating this. Saves me trying to explain it in words so much.
This concept of trajectories applies to all celestial bodies, and space travel between them. The Apollo missions to the Moon didn’t involve “swimming” in space from the Earth to the Moon, but rather first orbiting the Earth, and then thrusting prograde (using rockets to propel in the forward direction of the orbit) in order to change the craft’s orbit so that its highest point from the Earth would meet the Moon’s sphere of influence (where the Moon’s gravity becomes the dominant gravitational force). This is called a Trans-Lunar Injection.
“So can you ever get left?”
It’s matter of being so far away that something other than the sun (the black hole at the centre of the galaxy) will exert a greater gravitational influence than the sun. I’ve read this to be approximately 1 light year away, or about 6 trillion miles. Anything nearer and you’ll still be orbiting the sun. Farther out, and you’re orbiting the black hole in the middle of the galaxy. This would be the Sun’s Hill Sphere.