You're right. The first stage (4 conical shaped boosters hanging off the sides) and the second stage (central core booster) all fire together to lift-off. When the side-boosters are depleted they jettison (and perform the Korolev cross) while the second stage keeps firing.

The third stage also starts before the second stage is completely depleted; this is why there's an "open grate" visible in photos of the rocket (about half way down the stack, see [1]), gives the exhaust somewhere to go before the previous stage is jettisoned. Hot staging is advantageous as you don't need extra ullage motors to deal with propellant being in the wrong place inside the tanks in microgravity. The Soyuz (and other R-7 variants) are brilliantly engineered; There's a reason they're still flying nearly 70 years on.

[1] https://upload.wikimedia.org/wikipedia/commons/thumb/9/9a/So...