Besides a couple thruster firings it's almost all coasting with the engines off after those first ~9 minutes of launch, and the coasting is why you feel weightless!

Mostly that time is spent hanging out in a lower/faster orbit, "catching up," which is needed because the station rarely passes exactly over the launch site at exactly the right time. This extra wiggle room gives them enough launch windows per day.

Scott Manley has a great video: https://www.youtube.com/watch?v=bUi0yWc5Dnw