You have code that runs two motors at the same time when the switch is pressed, in one direction, and in the other direction when the switch is not pressed. If you want one motor to start after the other, delay() is the simplest.
If you want to avoid delay, you need to think about a state machine. You have 4 states - one motor running forward, two motors running forward, one motor running backwards, and two motors running backwards. Depending on the state of the switch, you transition to one of two states. Depending on how long it has been since the last transition, you might need to transition to one of the other two states.
Finally, depending on the state, you activate certain motors certain ways.