Did we ever witness Kirk's ship launch directly to anything higher than warp two?
Yes. Although I think that the ship accelerates through the other warp factors. So if Kirk says, "Ahead, Warp 6" then they accelerate through 1,2,3,4, and 5. I would think TNG does the same thing since IIRC, they mention an "acceleration curve".
"Tomorrow Is Yesterday" - Enterprise accelerates to Warp 3 upon leaving orbit (and then continues on accelerating.)
KIRK: Ahead warp factor three, Mister Sulu.
"Errand of Mercy" - Enterprise accelerates to Warp 7 from what sounds like after they stopped at the "designated position".
KIRK: Lay in a course for Organia, Mister Sulu.
SULU: Aye, aye, sir.
SPOCK: Negotiating with the Organians will be time-consuming, Captain, and time is one thing we'll have the least of.
KIRK: We won't get it by talking about it. The trigger's been pulled. We have to get there before the hammer falls. Ahead warp factor seven.
"The Deadly Years" - Kirk orders Warp 8, but I'm unsure whether Enterprise's power loss slowed it down from Warp 5 to sublight or if they were still at Warp 5.
KIRK: Kirk, commanding Enterprise, out. Mister Sulu, course one eight eight degrees, mark fourteen. Speed, warp factor eight. Stand by.
"The Immunity Syndrome" - after they blow up the amoeba Kirk orders Warp 5 once they pick up the shuttle.
KYLE: Prepare to receive shuttlecraft. Received shuttlecraft. Hangar doors closed. Hangar deck pressurising.
KIRK: Mister Chekov, lay in a course for Starbase Six. Ahead warp factor five. I'm still looking forward to a nice period of rest and relaxation on some lovely planet.
"For the World Is Hollow..." Kirk orders Warp 3 after engaging the missiles at sublight.