> In the TV shows and movies, they never seem to spend more than a few days at a time going anywhere at warp speed.
There are a few canon reasons for this:
1) Ignoring special episodes (and all of Star Trek Voyager) where they use a contrivance to hurl the ship tens of thousands of light-years, the working range of a starship on Star Trek is relatively small:
* The distance to the Romulan Neutral Zone, on the outer fringes of Federation Space, is only 28 light years away from Earth.
* Deep Space 9 (so named because it's on the outer frontiers of known space) is 70 light years away from Earth.
* Oo'nos, the Klingon homeworld, is ~100 light years away from Earth.
2) Ship speed exponentially increases as warp factor increases, and by the time the USS Enterprise-D is in service, warp 9 is fast, but not "I need to get there immediately because every plot shown on TV is an emergency" fast. The maximum sustainable speed for the Enteprise-D is warp 9.6, which is 1,909 times the speed of light. This would allow Picard to reach even Qo'nos from Earth in under 3 weeks.
3) With the exception of the JJ Abrams franchise (where they seem to keep going back to Earth after every mission), the ships featured in Star Trek are generally exploratory: between each episode, they'd roam their mission areas performing routine work. The episode plots would generally revolve around them dealing with an issue that happened to be close to wherever they happened to be at the time.
Of course, being a set of TV series and movies, liberties with distances were routinely taken. For example, in the first episode of Enterprise, they say Qo'nos is 4 days away from Earth at warp 4.5, which would place it less than half the distance between Earth and Alpha Centauri, the closest star system.
Most of the time they're moving at the speed of plot.
It's a log scale I think so warp 9.9 is much faster than warp 9. Except whenever they remember the terrible idea they had part way into The Next Generation when they put a speed limit on space travel.
The TNG scale have a asymptote at Warp 10. The TNG scale come from a draw, and actually have a very complex formula derive from these hand draw graph, but can be approximated to a exponential scale below Warp 9.
TOS (and Enterprise) uses other scale with a much more simpler math exponential formula, were the same Warp factor is less quick that in TNG scale.
Actually the TNG scale is because Rodenbery like to give the impression that the ships are more fast that in TOS but he like to avoid using Warp 10 and huger numbers.
In the TV shows and movies, they never seem to spend more than a few days at a time going anywhere at warp speed.