Depends on the technology involved.
A fluid clutch acts as a torque multiplier, which helps with launches if traction isn't an issue. However the fluid clutch consumes power, so it's a handicap in higher gears. Some fluid automatics use a mechanical clutch for top gear to avoid this loss and get better gas milage.
The shift times on an automatic can be made very quick. For "classic" automatic transmissions, a "valve body" which is fluid driven mechanical logic, controls the shift time and when shifts occur. For a passenger car, shifts are made smooth by having two gears partially engaged at the same time with a smooth transition. For a drag racing car, there's no need for smooth shifts, so the overlap period is made very small, and the required rpm to launch is set higher.
A mechanical clutch is a torque limiter, converting power to heat while it slips. On a high powered car, the limiting factor is tire traction, not engine torque so it doesn't matter. Humans can only shift so fast, but road racing cars can use electoncially programmed no lift sequential shifters, like XTRAC, which can do shifts in 50ms to 30 ms, depending on the gear (rpm drop is a factor), with the driver just toggling butons or a shift switch.
DCT's (dual clutch transmissions) use two sets of mechancal clutches, one for odd gears, the other for even gears. These also use electronic logic to shift gears. Some of these also include launch control for the mechanical clutch, so the driver never needs to use a clutch pedal.
Formula 1 doesn't allow DCT's, but they get the equivalent effect by having the clutch and gearbox controlled by the same ECU that controls the engine. Gearshifts involve a short period where both gears are engaged at the same time (the taller gear takes the load on an upshift), and the clutch and fuel are controlled for a smooth shift while rpms quickly change to prevent tire slippage and protect the drive train. There were 1 or 2 years where Formula 1 race cars were fully automatic, including launch control (clutches were used in pit stops).
Top fuel drag cars only have a single forward gear (plus a reverse), and the clutch is mechanically programmed with springs and weights to try an keep the tires at the limits. The crew has to estimate how much traction there is on a run when programming how much initial torque to use and quickly the clutch increases torque during a run (the cars have downforce, so the torque is increased during the first part of a run until the clutch is fully engaged).
A Corvette Z06 goes 60 mph in 1st gear, with a top speed of 198 mph in 5th (but it would take 3 to 4 miles to reach that speed). 6th gear is a true overdrive. Some exotic cars are geared even taller. The fastest bikes like a Suzuki Haybusa or Kawasaki ZX14 reach just over 80mph in 1st gear. Top speed is 190+ mph, but there's a speed limiter set to 186 mph (300 kph).
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