The times we are given some good distance / time information in TOS was "Obsession" and "That Which Survives" and that pretty much tells us that going between star systems the Enterprise can easily do 1,000 LY in a day. TOS speeds were generally faster when it came to interstellar travel so going to the galactic edge wouldn't have taken too long.
TNG slowed things down a bit and the dialogue puts the E-D at 833c @Warp 9 in "Bloodlines". Voyager sped things up a bit and was often cruising above Warp 9. According to "The 37's", Warp 9.9 was 21,457c.