In the TNG episode "The Nth Degree," Barcley uses a subspace inversion to travel the enterprise a phenomenal distance, but it doesn't seem to be an actual mode of transportation. Maybe this is a one-off fastest form of travel. Are transwarp conduits the fastest consistent form of travel? So my question is what is the the fastest one-off or special form of travel, and what is the fastest standard or widely used form of travel? And what rate of speed are we talking about?
In Voyager "Threshold", Tom Paris achieves Warp 10: "infinite velocity"
KIM: In principle, if you were ever to reach warp ten, you'd be travelling at infinite velocity.
NEELIX: Infinite velocity. Got it. So that means very fast.
PARIS: It means that you would occupy every point in the universe simultaneously. In theory, you could go any place in the wink of an eye. Time and distance would have no meaning.
In the episode, Tom Paris actually achieves this speed via a kind of "transwarp" technology developed for one of Voyager's shuttles, resulting in weird consequences for the character and one of the Star Trek franchise's most questionable products:
As you can't get faster than infinite velocity and occupying all points in space at once, this is the fastest one-off form of travel in Star Trek.
The fastest standard / wisely-used forms of travel that we have seen on-screen would be Borg transwarp conduits: while less than Warp 10, they greatly exceed Voyager's top speed of Warp 9.9 given what they allow Voyager to achieve in the final episode, "Endgame".
Quoting from the Star Trek TNG Technical Manual (Considered a canon source of information about the Trek universe), we have the following "author's note" from series Writer and Producer Rick Berman
"Figuring out how "fast" various warp speeds are was pretty complicated, but not just from a "scientific" viewpoint. First, we had to satisfy the general fan expectation that the new ship was significantly faster than the original. Second, we had to work with Gene's recalibration, which put Warp 10 at the absolute top of the scale. These first two constraints are fairly simple, but we quickly discovered that it was easy to make warp speeds TOO fast. Beyond a certain speed, we found that the ship would be able to cross the entire galaxy within a matter of just a few months. (Having the ship too fast would make the galaxy too small a place for the Star Trek format.) Finally, we had to provide some loophole for various powerful aliens like Q, who have a knack for tossing the ship millions of light years in the time of a commercial break. Our solution was to redraw the warp curve so that the exponent of the warp factor increases gradually, then sharply as you approach Warp 10. At Warp 10, the exponent (and the speed) would be infinite, so you could never reach this value. (Mike used an Excel spreadsheet to calculate the speeds and times.) This lets Q and his friends have fun in the 9.9999+ range, but also lets our ship travel slowly enough to keep the galaxy a big place, and meets the other criteria. (By the way, we estimate that in "Where No One Has Gone Before" the Traveler was probably propelling the Enterprise at about Warp 9.9999999996. Good thing they were in the carpool lane.)"
So there you have it. The Q have the ability to travel (within the bounds of the universe) at speeds of warp 9.9999+. Not only that, but as we can see from the Voyager episode Death Wish, they're also able to use time travel to travel to any point within the universe's existence, effectively rendering them capable of traveling to any point in the universe instantaneously.
So, the fastest way to travel is...by clicking your fingers and wishing.
Wormholes in the Trek universe allow for transit to any linked point in the galaxy instantly. In fact, the linked point might actually be at a previous time as well (which I think makes any "velocity" calculation you might try to perform involve either a negative result or an imaginary number, depending on how you look at it).
There was a relatively stable wormhole that was used for transit purposes in the DS9 series. According to Daniel in the comments below, traveling through it works out to about Warp 9.999999999999916.
Now you could argue that any "velocity" number derived for wormhole travel is nonsensical because you aren't transiting all the points in between your two points. However, that same argument could be made for any supposed velocity past Warp 1. You aren't actually traveling faster than light (because that can't be done), but rather you are cheating the universal speed limit by playing tricks with space-time.
A Repulsor Beam
Perhaps not the fastest, but a VERY fast form of propulsion was that used by the Aldeans. They managed to transport the Enterprise-D 3 days away at warp 9 almost instantaneously:
RADUE [on viewscreen]: Captain, we want you to understand the nature of your choice. A small demonstration of our power. (Something whizzes around the planet and knocks the Enterprise spinning out of orbit, into deep space)
PICARD: What was that?
DATA: I believe it was a repulsor beam.
PICARD: Position report.
LAFORGE: This is unbelievable, sir. According to my calculations, we're three days from Aldea. At warp nine.
RIKER: And they call that a small demonstration?
PICARD: Geordi, get us back to Aldea. Warp nine.
LAFORGE: Aye, sir.
RADUE [on viewscreen]: Captain, if you don't accept our terms, the Enterprise will be pushed so far away that by the time you return, your children will be grandparents.
(TNG 'When the Bough Breaks')
After doing some calculations, this is a speed of approximately 3.93 x10^12km/s (assuming it took them about 3 seconds to travel that distance) i.e. 3,930,000,000,000 km per second
Whether or not this was a 'small demonstration' or not is up for debate, but considering their technological feats of being able to cloak an entire planet, it wouldn't at all surprise me if they could push them that far away.
This is pretty fast indeed!
If I remember correctly, That Which Survives was a Star Trek TOS episode in which the Enterprise's warp engines rev out of control, and they pass Warp 13.2. This episode includes a 990.7 light-year leap, almost instantaneously. This has got to be fast.
Also, In 2364, the Traveler used the energy of his thoughts to move the USS Enterprise-D through space at a speed that registered on instruments as exceeding warp factor 10 and going off the warp scale. (TNG: "Where No One Has Gone Before")
For a technical means, and not supernatural Q/Traveler...
In the Star Trek: Destiny trilogy of novels (about 16 months after events of Nemesis), the highly advanced and xenophobic Caeliar created these subspace tunnels that were much faster than transwarp conduits and could go much further. One of them took the Enterprise to a proto-galaxy created just after the Big Bang—some 13-14 billion light years away— in a matter of seconds.
Don't know if this trilogy is considered canon or not.
Iconian gateways allowed "instantaneous" travel over distances of at least 70,000 light years.
In Star Trek Online, we find an Iconian gateway in the Andromeda Galaxy and that Iconian spacecraft open such gateways to their destinations, rather then putter along using Warp Drive.
This seems to be another "infinitely fast" form of travel.
Perhaps 31st Century Time Travel
I'm not going to get into the temporal side of it - yes any time travel means that you technically arrive before you left so you have negative speed...
I'm just going to restrict this to actual spatial travel.
In ENT 'Carpenter Street', Archer and T'Pol are sent back to 21st century Earth instantaneously by stepping through a door. Remember that the Enterprise was at this stage in the Delphic Expanse. According to this answer, the Delphic Expanse is about 50 light years from Earth, so they just travelled 50 light years in about a second. This sounds pretty close to Warp 10 (i.e. infinite velocity) to me!
In the episode Contagion, the Ikonians used gateways to instantaneously transport from one place to another, anywhere in the known universe--as easily as walking across a room. They were known as "demons of Air and Darkness."