Kepler's Three Laws of Planetary Motion are particularly helpful when addressing this question. They state that (in informal language)
- The shape of a planet's orbit in an ellipse, with the Sun at one focus of the ellipse.
- As planets move around their elliptical orbits, the imaginary line drawn from the planet to the Sun sweeps out equal regions of equal area in equal amounts of time.
- The square of the period of a planets orbit, $T^2$ is equal to the cube of planet's orbit's semi-major axis (a^3)
Although not immediately obvious, Laws 2 and 3 combined both imply that as a satellite (planet, asteroid, comet, or otherwise) approaches closer to the sun, it can be expected to have a faster velocity.
Specifically, if we look just at the eight planets, and Law 3, $$T^2\propto a^3$$ which, when solved for period states that $$T\propto \sqrt{a^3}$$ So using the equation above, let's say planet $A$ travels in some orbit around the sun, and the semi-major axis has a length of $a$. If planet $B$ travels in and orbit, with a semi-major axis of $4a$ then the period has now increased by a factor of 8, even though the semimajor axis (and approximately the circumference, if the orbit has an eccentricity close to 0) only increased by a factor of 4. So as you move away from the sun, your period increases more than your distance, which means your orbital velocity is decreasing. Just look at this graph below, taken from enchantedlearning.com.

You can see a clear relationship between velocity, and distance away from the sun.
Now let's look at interlopers to our solar system, like comets. Compared to planets, most comets tend to have eccentricities very close to 1 (which means their orbits are very elliptical). Some comets even have eccentricities greater than one, which means they're on one-time hyperbolic orbits around the sun. As these comets approach perihelion (the close approach to the Sun) Kepler's Second Law tells us that thevelocity of the satellite increases. The most extreme examples are sun-grazing comets, which have very close approaches to the Sun. In fact, comet ISON was moving so quickly last November when it approached perihelion that had a) you been able to see the comet in daylight and b) Coment ISON not met an untimely demise you would have actually seen it change position in the sky (relative to background starts) by the hour.