There is a quite common confusion, between the apparent movement of the Sun "through" the constellations and it's actual movement around the Galaxy. The two are entirely different. The illustration at the right shows how a "line of sight" towards the Sun makes it appear (from our viewpoint) as if it is moving along the Ecliptic, but this is still very much out of scale, and may not give a real feel for what is actually moving.
It is not easy to model 3 dimensional celestial movements in one's head, so the following analogy might help. It may seem rather silly, but sometimes the sillier the analogy, the better it sticks in the mind.
Imagine you are looking at a tall cactus, in a large flat desert, surrounded by distant mountain ranges. The cactus represents the Sun, you are the Earth, and the distant mountains are the Constellations of the Zodiac.
The cactus will appear to be lined up with one of the mountains. Now walk in a circle around the cactus, always looking towards it - you will see the cactus line up with different mountains. Notice that the cactus itself does not move any closer to the mountains. It is YOUR movement around the cactus that causes its VISUAL position against the mountains to change. Similarly, Earth's orbit around the Sun only causes the Sun's VISUAL position against distant constellations to change. This is what's meant when we say the Sun "passes through the Constellations of the Zodiac" or "passes around the Ecliptic" once per year. It is not a physical movement through them. It has nothing whatsoever to do with the Sun's orbit around the galaxy.
If we scale these distances to the same ratios as Earth, Sun, and the nearest star:- standing 10 metres from the cactus, the nearest mountain would need to be approximately 2,800 km away!
To add sunrise and sunset to the analogy, you would need to sit in a revolving office chair holding a video camera while someone pushes you around the cactus (I did say it would get silly :) - In your movie, your chair's spin would make the cactus come in and out of view (analogous to Earth's spin causing Sunrise and Sunset) and your being pushed around in an "orbit" would make successive cactus-rises happen "in" different mountains (analogous to the Earth's orbit around the Sun during 12 months). You could also tilt the chair at about 23 degrees to simulate earth's axial tilt.
You could name some groups of mountains after the shapes they appear to make, for example in the picture above there's a region that looks like a Camel's humps, but again this is only a line-of-sight effect, the individual mountains may not be physically associated, and would not form that shape if viewed from somewhere else. When we hear Astrologers say things like "the Sun is in Capricorn", this is just as irrelevant and meaningless as saying "the Cactus is in the Camel".
There have also been claims (see the Earthplay page) that changes in the ellipticity of Earth's orbit moves us closer to different constellations. It doesn't:
Imagine your "orbit" around the cactus changes from a circle to a slight ellipse. Does this move you significantly closer to any of the mountains (which are at a scale distance of at least 2,800km)? Of course not. Just as changes in Earth's orbital eccentricity do not move it closer to constellations.
So we have seen that the Sun's annual passage "through" the constellations is illusory, and is not due to it's orbit around the galaxy. However on very much longer timescales the Sun and the Solar System do move relative to the rest of the galaxy, and this will slowly change the appearance of constellations, but again this motion should not be confused with the orbital speed of the Sun around the galaxy. The RELATIVE motion of an object within the rotating reference frame of the galaxy is called it's "Proper Motion". In the case of the Sun it is known as the Solar Apex (or "Apex of the Sun's way") and is in the general direction of the star Vega at about 16.5 km/s.
An analogy often given to explain this is horses on a carousel: The horses all "orbit" the carousel together, without moving towards or away from each other. If a horse did possess some movement towards or away from another horse, then that would be called it's "Proper Motion" and would NOT include the general rotational speed of the carousel. Similarly, in the case of the Sun, it's 16.5 km/s Proper Motion towards Vega does NOT include it's 220 km/s motion around the galaxy.
Note on scale calculation:
10 metres * (distance to Alpha Centauri) / (distance to Sun)
10 metres * ((4.2 * 10^13 km) / 150,000,000 km)
P.S. Those mountains only exist in Photoshop.