Después = Later Después de = After Luego = Then

you can use both with the same meaning... but if we put some examples: "despues" si more used as After Despues de comer = After the dinner Luego is rarely used in my country but the meaning is nearly the same, is more about the context, like news: Luego de tener una discución la esposa se fue de la casa = After having a quarrel the wife left the house "Luego" is more used in reported speach...

Después is closer to after, luego is closer to later.

thouse words mean the same thing. If you are not sure use Después.