Star trek ships have to “plot a course”; that implies that they use star charts to calculate route. It might also explain inconsistencies in speed – some parts of galaxy are more hazardous for navigation than other, plus in unexplored space they will have to use sensors to evade hazards (stars, planets etc) so that might be one of factors contributing to awfully low speeds in Star Trek : Voyager (75 years for 70 000 light years or 933 c) when compared to some other episodes. It is also possible that Voyager's usage of Bussard collectors to partially resupply during warp flight dictaded extremely low speed when compared to longer-range Galaxy class.
For example, in episode "The Chase" we have route that Professor Galen was expected to finish in few days to 6 weeks in shuttle.
Our galaxy is light years in diameter, and route shown is approximately equal to 4/5ths of its radius, or 40 000 light years. Most TNG shuttles are able to achieve warp 6, so we have warp 6 being 347 857 c in explored space.
According to TNG warp formula table here, warp 9 is 3.867 times faster than warp 6. That would give us warp 9 being 1 345 283.7 c in explored space and possibly under ideal conditions. That is why I tend to use Graham's idea of "warp highways" where ships travel faster than under "normal" conditions, where speeds are denoted by warp formula. Also, it is mentioned that usage of high warp "overextended" warp drive.
"Normal" warp speeds:
http://memory-alpha.org/wiki/Warp_factor
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TNG, "Encounter at Farpoint":
At this time, top speed for the Enterprise-D is about warp 9.3. Warp 9.8 may be possible, but it may also destroy the ship.
TNG, "Where None Have Gone Before":
Maximum warp speed is about 9000c. (Data reports the distance as 2.7 million ly, and Geordi reports a 300+ year travel time.) As of the pilot, maximum warp is about 9.3.
TNG, "The Icarus Factor":
Riker claims that it would take months at high warp to get from the current location of the Enterprise to the Vega-Omicron sector, on the far side of Federation territory. Assuming the upper limit on distance is 8000 ly (per Star Trek: First Contact) and the lower limit on travel time is two months, the upper limit on speed is 48,000c. Actual speed is probably far lower.
TNG, "Q Who":
Data reports they are 7000 ly from Federation space, and it would take them 2 years and 7 months of travel to get there. This indicates that the maximum sustainable speed of the Enterprise-D is about 2700c.
TNG, "Up the Long Ladder":
The trip from the Bringloid system to NB2323 is 0.5 light-years and takes more than 18 hours at warp 5. This means that warp 5 is less than 244c.
TNG, "Bloodlines":
It would take 20 minutes to travel 300 billion km at warp 9; this would put warp 9 at 833c. This estimate is an outlier, being quite low compared to other TNG speed examples.
TNG, "Clues":
0.54 parsecs is about a day's travel at cruising speed. This would indicate a cruising speed of 644c.
TNG, "The Most Toys":
Kivas Fajo's merchant ship, with a maximum speed of warp 3, could have traveled no more than 0.102 light-years in 23 hours. This is about 39c.
VOY, "The Caretaker":
According to Janeway, at "maximum warp", Voyager's 70,000+ ly trip back to the Federation would take 75 years. This is about 1,000c. "Maximum warp" may represent the best speed they could sustain over that distance with their limited resources.
VOY, "The 37's":
According to Tom Paris, warp 9.9 is 4 billion miles per second, which is 21,473c.
VOY, "Unimatrix Zero":
Voyager travels 2 light years in 2 hours when responding to a distress call. This is about 8,766c.
VOY, "Maneuvers":
Harry Kim gives a current speed of 2 billion kilometers per second, about 6,667c.
VOY, "Scorpion":
Chakotay states that at Voyager's maximum warp, a 40 ly trip would take five days which is just under 3,000c.
VOY, "Hope and Fear":
It takes Voyager two days at high warp to travel 15 Light-years. This is approximately 2,740c.
VOY, "Equinox""
Using an exotic fuel, the USS Equinox supposedly traveled 10,000 ly in about 2 weeks, for a speed of 260,893c. It should be noted, however, that there is no actual evidence that USS Equinox achieved such speeds other than a statement from Captain Ransom, a known liar and murderer.
VOY, "Friendship One":
A 132 light-year side-trip to investigate a planet would require two months round-trip at "maximum warp". This equates to a speed of about 1584c.
VOY, "Critical Care":
When Janeway asks how quickly Paris can get Voyager to an asteroid 3 ly away, Paris replies 2 hours. This is approximately 13,149c. The "How fast can you get us there?" question suggests top speed.
ENT, "The Expanse":
The Enterprise (NX-01) took seven weeks to reach the Delphic Expanse. That is approximately 373c.
ENT, "Damage":
The Enterprise (NX-01) travelled 4 Light Years in 3 days. That is approximately 487c.
ENT, "Detained":
The Enterpise (NX-01) traveled 5.2 Light-years in 3 days. That is approximately 633c.
ENT, "Affliction":
The Enterprise (NX-01) has become capable of Warp 5.2.
TOS, "That Which Survives":
Spock predicts that the Enterprise can travel 990.7 ly in 11.33 hours at warp 8.4, for a speed of 766,503c.
TOS, "Obsession"
The Enterprise pursues an alien cloud creature to it's point of origin 1,000 light years away in the Tycho system. Kirk then notes that they must make a rendezvous with the starship Yorktown, coming back the 1,000 ly to make the rendezvous in 48 hours. Thus one day to get to the Tycho system and one day to come back for a speed of 365,000c
TOS, "The Squire of Gothos"
The Enterprise is 900 light-years from Earth. While this is not a specific speed versus distance quote, it does show that the Enterprise can travel in one week out into unexplored territory across nearly a thousand light years, then the next week be somewhere completely different for the events of "Arena".
TOS, "The Cage/The Menagerie"
Pike attempting to communicate with his Talosian captors "My name is Christopher Pike, commander of the space vehicle Enterprise from a stellar group at the other end of this galaxy". This implies travel on a galactic scale, and would require speed and knowledge unlike anything seen in all of Trek. Some have taken it to mean not the literal other opposite side of the Milky Way galaxy from where Earth would be located, but something much closer.
TOS, "Where No Man Has Gone Before"
The Enterprise seems capable of routine travel to the very literal edges of the galaxy where they are turned back by a mysterious energy barrier. This will not be the only time the ship travels this far and beyond in TOS; "By Any Other Name" and "Is There In Truth No Beauty?". Even if the ship is somehow heading "up" or "down" straight out of the Galaxy though the shortest distance possible, not going to the outer edges of the spiral arm disk, it still means the Enterprise has to traverse across thousands of light years, and all in a very short span of time.
TOS, "Bread and Circuses"
The Enterprise travels across one-sixteenth of a parsec in "seconds". This would equate to a speed of approximately 215,000c, assuming it took a full 30 seconds to get to the system, never mind the specific planet there.
Trekkies have difficulty explaining why Federation starships of the 24th century would be substantially slower than Federation starships of the 23rd century. Their response is that Warsies cherrypick modern Trek for the lowest showings, ignoring higher examples, such as the following:
TNG, "The Chase"
Picard's old professor Galen marks out a path on a map of the Milky Way galaxy that must be followed in order to put together an ancient puzzle. The line Galen traces subtends an impressive 40,000 lys. Galen then notes that with a shuttle or transport, the trip will require months to complete, but if he had access to a starship, then it would only be a matter of weeks. The minimum speed a starship would require to complete the journey as shown would require an average speed of 1 million c.
TNG, "The Best of Both Worlds, Part I and II"
The E-D chases the Borg cube back to Earth over the course of 6 days from the edges of Federation territory. Assuming the nearest edge of Federation space was just 1,000 ly away from Earth, it would require the E-D to maintain a speed of nearly 61,000 c.
DS9, "Valiant"
The cadets tell how the original mission of the Defiant-class USS Valiant was to circumnavigate the 8,000 ly Federation in 90 days. Assuming by that they ment the ship was to go straight out 8,000 lys and then come back to Earth with no stops along the way, then the ship would need a speed of 64,888c.
However, if "circumnavigate" means "circumvent", we have 2 x 4000 ly x pi or 25 000 light years in 90 days. That is 102 000 c.
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Warp manouvering
Janeway: "Tom, what's the first thing they teach you about maneuvering at warp?"
Tom: "'Faster than light, no left or right.' When possible, maintain a linear trajectory. Course corrections could fracture the hull."
Janeway: "Exactly. We'd have to drop to impulse every time we made a course change."
That basically throws any possibility of radical warp manouvers out through window (it also explains why Dominion fleet was able to simply wait for Federation fleet to come to them - "hold a roadblock", as I put it on Factpile. Federation fleet did not want to lose time by taking longer route, and Dominion fleet blocked route they were going - after all, Dominion has sensors capable of detecting ships from way longer range than Federation can).
There are some examples of warp manouvering, as seen here. However, all these examples involve one to two ships at most; performing warp manouvers with entire fleet will be extremely difficult at best, especially if fleet has large number of older ships. Also, in TOS episode "The Ultimate Computer" we have clear and relatively rapid warp manouvering at warp 4.
On my site.
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