I would think you'd have to instantaneously accelerate because incremental acceleration doesn't work the way we typically think it does at high speeds.

If you're moving at 99.999% the speed of light relative to Earth, anything close to your speed is going to be moving quite slowly relative to you. When you accelerate some more, the change in speed relative to those close things is much larger than the change in speed you experience relative to Earth (it gets smaller and smaller as you approach light speed). But as far as I understand, there's no such thing as moving at light speed relative to Earth but not relative to other sub-light speed things. You'd have to instantaneously move at light speed relative to everything (every sub-light speed thing).