The right-back signed a five-year contract to take the Ibrox club's close-season spending above £7m.
Whittaker said it was "the right time" to leave Easter Road. Now he hopes to secure a regular place in Walter Smith's starting XI.
"With the size of a club like this, there's always going to be competition," he said. "I need to work hard on my game but hopefully I can force my way in and then stay in there.
"I feel I attack better than I defend, and I think it's been said that attacking is the strongest point in my game.
"I've been thinking about making his debut a lot, playing in front of the home crowd. It's something that people dream of when they were younger and I'm no different."