Each new star we find is generally considered to be part of the constellation it is nearest to.
Our Sun is obviously a star, just much closer. Is our Sun part of any constellation? If so, which constellation is it a part of?
Constellations are human constructs to make sense of the night sky. When you are trying to find your way around, it helps to "chunk" stars into patterns and assign those groupings names. When I want to point out a particular object in the sky (say Polaris, the North Star), I start by pointing out a familiar constellation (say Ursa Major, the Big Dipper). From there, I can tell my friend to follow this or that line to get them to look where I'm looking:
With the advent of computerized telescopes and large data sets, constellations are less important for professional astronomers. However, many stellar databases use Flamsteed or Bayer designations, which assign stars to constellations. In order to include all stars, the sky is divided into irregular regions that encompass the familiar constellations.
So, which constellations is the Sun assigned to? Well, from the perspective of someone on the Earth, the Sun moves through the constellations throughout the course of the year. Or rather, Sol moves through the region of the sky where some of the constellations would be seen if its light did not drown out distant stars. Our moon and the rest of the planets move through those same constellations. (The Greek phrase which gives us the word "planet" means "wandering star".)
The current position of the sun against the background of distant stars changes over the course of the year. (This is important for astrology.) It's a little easier to make sense of with a diagram:
So perhaps a better question is:
Presumably an observer on an exoplanet would assign Sol to some constellation that is convenient from her perspective. But from our perspective within the Solar system our sun, moon, and planets are not part of any constellation.
No, it does not. The constellations are fixed (on time scales long enough for humans to consider as fixed, at least) patterns of stars which exist on the celestial sphere. This celestial sphere is a coordinate system which has the Earth at its center. From the Earth's perspective, the sun rises and sets at the same rate as the constellations, but as the Earth revolves around the sun, the constellation the sun will appear to be in will change.
However, the constellation the sun appears to be in (if projected onto the celestial sphere), is exactly how the constellations of the Zodiac are determined.