If the MFDU is set up correctly, remember the autopilot Flight Control Unit must be as well.
It should show a series of dashes under speed, then a dot, dashes under heading, then a dot, your selected altitude, then a dot.
The dashes indicate the MCDU is controlling the value.
The dots indicate a function has been selected for AP control.
Engage autothrottle prior to T/O.
Advance throttles to TOGA setting.
A/P can be turned on shortly after positive rate of climb established, at acceleration altitude, reduce the
throttles to the climb detent.
You do not need to touch any other buttons, aside from entering new altitudes and re-engaging the climb as per ATC, basically until you are on approach.
This link is to the documentation for the FBW A320, although more advanced than the default aircraft, a lot of the info is still applicable and covers these subjects.
Good luck!