Autopilot in once the flight path is under control and you are above 250ft
I would 'Prefer' LNAV & VNAV.. In most cases it is already armed on the ground so there is no need to change the modes. You may need HDG SEL if there is a special Eng out procedure otherwise LNAV will give appropriate AOB protection and will fly the SID so why not use it.
VNAV is also an appropriate and intelligent mode as it will give you accelleration to Vref30+100 and it will give you CON thrust once cleaned up and at vref+98. Ie it will disregard the Thrust reduction alt/flaps set in the FMC if it has detected an engine failure.
another option if VNAV isn't used is FLCH or even Stay in TOGA. in both those cases you will need to set vref+100 manually you will also need to Push the THR button once cleaned up and at vref+100
In the Go around case. If pressing vnav at accel height then it will immediately set CON thrust but only if ENG OUT prompt was selected .. otherwise if ENG OUT isn't selected then it will set CLB thrust. when using FLCH or TOGA for GA it is the same as the Take off scenario above